Optimal Grain Mixing is NP-Complete

TL;DR

This paper proves that finding the optimal grain mixing strategy for maximizing profit in wheat transportation is computationally hard, specifically NP-Complete, through complexity reductions.

Contribution

It provides the first complexity proofs showing the grain mixing problem is NP-Complete, indicating the problem's computational intractability.

Findings

Proves the grain mixing problem is NP-Complete.

Reduces the problem from 3-dimensional matching and planar 3-DM.

Exact solutions are likely infeasible due to complexity.

Abstract

Protein content in wheat plays a significant role when determining the price of wheat production. The Grain mixing problem aims to find the optimal bin pair combination with an appropriate mixing ratio to load each truck that will yield a maximum profit when sold to a set of local grain elevators. In this paper, we presented two complexity proofs for the grain mixing problem and showed that finding the optimal solutions for the grain mixing problem remains hard. These proofs follow a reduction from the $3$-dimensional matching ($3$-DM) problem and a more restricted version of the $3$-DM known as planar $3$-DM problem respectively. The complexity proofs do suggest that the exact algorithm to find the optimal solution for the grain mixing problem may be infeasible.