The average shadow price for MILPs with Integral resource availability and its relationship to the marginal unit shadow price
Abstract
The economic significance of the average shadow price for integer and mixed integer linear programming (MILP)
problems has been established by researchers [Kim and Cho, Eur. J. Operat. Res. 37 (1988) 328; Crema Eur. J. Operat.
Res. 85 (1995) 625]. In this paper we introduce a valid shadow price (ASPIRA) for integer programs where the righthand
side resource availability can only be varied in discrete steps. We also introduce the concept of marginal unit shadow
price (MUSP). We show that for integer programs, a sufficient condition for the marginal unit shadow price to
equal the average shadow price is that the Law of Diminishing Returns should hold. The polyhedral structures that
will guarantee this equivalence have been explored. Identification of the problem classes for which the equivalence holds
complements the existing procedure for determining shadow price for such integer programs. The concepts of ASPIRA
and MUSP introduced in this paper can play a vital role in resource acquisition plans and in defining efficient market
clearing prices in the presence of indivisibilities.
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