Robust minimum cost flow problem under consistent flow constraints

Abstract

The robust minimum cost flow problem under consistent flow constraints (RobMCF$$\equiv $$ ≡ ) is a new extension of the minimum cost flow (MCF) problem. In the RobMCF$$\equiv $$ ≡ problem, we consider demand and supply that are subject to uncertainty. For all demand realizations, however, we require that the flow value on an arc needs to be equal if it is included in the predetermined arc set given. The objective is to find feasible flows that satisfy the equal flow requirements while minimizing the maximum occurring cost among all demand realizations. In the case of a finite discrete set of scenarios, we derive structural results which point out the differences with the polynomial time solvable MCF problem in networks with integral demands, supplies, and capacities. In particular, the Integral Flow Theorem of Dantzig and Fulkerson does not hold. For this reason, we require integral flows in the entire paper. We show that the RobMCF$$\equiv $$ ≡ problem is strongly $$\mathcal {NP}$$ NP -hard on acyclic digraphs by a reduction from the (3, B2)-Sat problem. Further, we demonstrate that the RobMCF$$\equiv $$ ≡ problem is weakly $$\mathcal {NP}$$ NP -hard on series-parallel digraphs by providing a reduction from Partition. If in addition the number of scenarios is constant, we propose a pseudo-polynomial algorithm based on dynamic programming. Finally, we present a special case on series-parallel digraphs for which we can solve the RobMCF$$\equiv $$ ≡ problem in polynomial time.

Robust transshipment problem under consistent flow constraints

“…Before concluding this section, we present some related results of the capacitated version of the RobT $$$ \equiv $$$ problem, the RobMCF $$$ \equiv $$$ problem, analyzed in our previous work [9]. We prove the ‐hardness of the RobMCF $$$ \equiv $$$ problem on acyclic, SP, and pearl digraphs.…”

“…The RobT $$$ \equiv $$$ problem is the uncapacitated version of the robust minimum cost flow problem under consistent flow constraints ( RobMCF $$$ \equiv $$$), introduced in our previous work [9]. The uncapacitated version is of interest, even if logistic applications might have a limited transshipment capacity, as the limitations are often sufficiently large.…”

“…On the basis of our previous work [9] we present the following minor contribution. First, we prove that deciding whether a feasible solution to the RobT $$$ \equiv $$$ problem exists is strongly ‐complete on acyclic digraphs, even if only two scenarios are considered that have the same unique source and unique sink.…”

Consumption Coupons, Consumption Probability and Inventory Optimization: An Improved Minimum-Cost Maximum-Flow Approach

“…Uncertainty theory is an efficient tool to deal with nondeterministic information, especially expert data and subjective estimation [37]. Scholars have conducted multidimensional studies based on different application scenarios [38][39][40], combined with the minimum-cost maximum-flow network model, and have achieved fruitful results.…”

Integrality in the multinetwork min‐cost equal‐flow problem

“…Morrison, Sauppe and Jacobson [15] show that any algorithm designed for the equal-flow problem can also be used to solve a generalization of it called the proportional network flow problem in which the flows on edges in given sets must all be proportional. Büsing, Koster and Schmitz [5] present an extensive survey of equal-flow related problems and their applications. In their model, costs and capacities are fixed but demands vary among network scenarios.…”