minimumcost

Minimumcost (minimum cost) refers to the lowest possible total cost achievable among feasible solutions to an optimization problem. It is a central objective in operations research, computer science, and economics, where the goal is to minimize expense while satisfying constraints such as capacity, demand, or precedence relations.

In graph theory, the minimum-cost flow problem asks for a feasible flow of a given value that minimizes the total cost, where each arc has a capacity and a cost per unit of flow. Solutions must satisfy flow conservation at intermediate nodes and respect capacities. Standard algorithms include the successive shortest augmenting path method, cycle-canceling, and the network simplex, sometimes with potentials to handle negative costs.

Related problems include the minimum-cost path, which seeks the cheapest route between two nodes, solvable by algorithms such as Dijkstra for nonnegative costs or Bellman-Ford for graphs with negative costs. The minimum-cost bipartite matching problem, solvable by the Hungarian algorithm or by reductions to minimum-cost flow, is a core example in assignment problems. Many problems combine these ideas in minimum-cost network design or production planning contexts.

Applications span logistics, supply chain design, telecommunications, energy systems, and manufacturing. Practical models often assume linear costs for tractability, though non-linear or stochastic costs may require more advanced approaches. The concept is also used in portfolio optimization, project selection, and budgeting to identify choices that minimize expense while meeting requirements.