Questions tagged [network-flow]

For questions related to the formulation or solution methodologies of network flow problems. Some of the well-known network flow problems are maximum flow problem, minimum-cost flow problem, and multi-commodity flow problem.

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Reformulating undirected to directed edges for MCF

As stated in this paper, there is a technique to reformulate a multi-commodity flow problem (MCF) with undirected edges to its equivalent version with directed edges. By quoting them: The ...
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6 votes
1 answer
331 views

How to Model Path Being Only Through Adjacent Cells

I am trying to develop a MILP for a path planning problem. I am operating on a grid of cells that represents a map. The arrangement of cells in the grid represents the placement of the cells in real ...
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-1 votes
1 answer
60 views

Assignment Problem with continuous decision variable

I have to solve a problem from industry where there are a number of machines which should be assigned to a number of tasks. The difference from the general assignment problem is tough, that the ...
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1 vote
1 answer
87 views

Help with dual of a problem

Could anyone confirm me if I write the correct dual for my problem? The different sets confuse me a lot. $s$ is the source node and $t$ the sink node. I'm uncertain if the last dual constraint in ...
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2 votes
1 answer
510 views

Model the TSP as a shortest path

We have a directed graph $G=(N,A)$ where $N$ are the verticies $N=0,1,2,3,4,5,..,n$ the starting location is node 0. For each arc $a=i-j$ in A we have a distance $d_a>0$ So we want to start and ...
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0 votes
1 answer
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How to model a simple routing problem using nodes and arcs

I have the following problem. I have a route which begins at node $0$. I want to begin and end in $0$ which is a depot. Now in this route even customer will be visited in order for example customer 2 ...
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1 vote
0 answers
68 views

How to see fix constraint violations found with IIS

I'm running a simple MCNF model to see if my network is balanced. I'm using Gurboi in Python and with the IIS I can see that there are 2 constraint violations making it infeasible. How do I fix these ...
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3 votes
4 answers
310 views

How to see constraint violation for network flow problem

I am testing to see if my network is balanced as part of a larger protection-interdiction-restoration problem. To do this, I'm solving the problem as a minimum cost network flow problem to see if the ...
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4 votes
2 answers
124 views

How to fix unbalanced multi-commodity network flow with equal supply and demand?

I have a fairly large network with eleven commodities and arc capacities that are commodity-dependent (i.e. an arc may have a higher capacity for one commodity than another). I'm solving a protection-...
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1 vote
0 answers
122 views

How to eliminate large matrix coefficients and large RHS warning in Gurobi

I'm solving a large network flow problem using Gurobi to do Benders decomposition in Python. The imported .csvs contain values that typically fall in either zero, the range of [20,200], or with a ...
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3 votes
2 answers
173 views

Designing a multi-commodity network flow optimizer

I'm trying to solve multi commodity multi source network flow optimization problem using Python-PuLP. Here is how my problem looks like: The numbers on the arcs represent the order of priority a ...
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2 votes
2 answers
195 views

Min cut network for patients

I'm trying to formulate a min cut algorithm There are two classes of patients: 1, 2, and the maximum number of patients that can be assigned to each bed per day is shown below as well. $$ \begin{array}...
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6 votes
2 answers
232 views

Multi-commodity Network Flow: Convert a node-arc solution to an arc-path solution

For a logistics optimization problem I am working on, the most natural expression of the problem is using a node-arc formulation of the MCNF. It is solved using an LP/MIP solver like CPLEX/Gurobi. ...
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3 votes
0 answers
102 views

Sources of Min-Cost Flow Models That Utilize Binary Variables for Transportation Networks

I am looking for articles that include min-cost flow models with binary variables for flow transportation like gas networks, traffic systems, heating systems. Is there any specific place(like OR ...
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  • 105
3 votes
0 answers
86 views

Availability of library for solving generalized network flow problems

I need to solve a large-scale generalized network flow problem. For reference, it is a pure network-flow model with the exception that arcs have a positive coefficient applied to flow on them. In my ...
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2 votes
1 answer
164 views

How to solve a minimum cost flow problem with time constraints

In the following picture, the decision-maker has to make deliveries to node $V=\{1,2,3,4,5,6\}$. He has two options: use cars or bikes. In both cases, he has to make a stop at a relay point where the ...
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  • 51
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0 answers
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Asserting solution existence via approximation of multi-commodity integral flow

Consider a polynomial $\epsilon$-approximation for MCF with integral flow (like the one based on LP relaxation). Can I (in general) use this algorithm as a predicate, asserting whether a solution ...
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3 votes
1 answer
76 views

Does multi-commodity flow problem (MCF) block cycling flows?

I am working with multi-commodity flow at integral flow with unit demand for each operation. Since I use the MCF formulation to model path finding, in my scenario cycles are just a waste of resources. ...
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5 votes
1 answer
103 views

(Integral) multi-commodity flow on undirected graph

Usually in (integral) multi-commodity flow problems the graph is assumed to be directed. Instead, I am working with an undirected graph. Is it possible to transform it in a digraph? Does such a ...
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1 vote
0 answers
32 views

Multi-commodity flow problem reduction

Assume to tackle a particular group of instances for the integral multi-commodity flow problem. Particularly, given a digraph $G = (V \cup S \cup T,E)$, such that $S, T, V$ are all disjoint. ...
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1 vote
0 answers
27 views

Solving a max-flow problem through lex-max flow problem

Assume that, for a given network, its multi-sink multi-source max-flow satisfies a lexicographic order. Is this enough to state that I can solve the max-flow problem, by using a lex-max solver (i.e. a ...
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1 vote
0 answers
21 views

Complexity of lexicographic integral flow

I read that the lexicographic flow problem is polynomially solvable. However it allows for continuous flow values. Is there any reference about the problem with integral flow (and its complexity)?
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1 vote
0 answers
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Is there a difference between "Integral Quickest Transshipment" and "Integral Quickest Multicommodity flow"?

I can't get the difference between this two problems, which seem equal to me. But, for the former, a polynomial solver is known, while, for the latter, it is not. EDIT - References: ...
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3 votes
1 answer
273 views

Is Multicommodity Flow problem really NP-hard?

Within paper ON THE COMPLEXITY OF TIMETABLE AND MULTI-COMMODITY FLOW PROBLEMS there's a proof that the multi-commodity is NP-hard. While in ...
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1 vote
2 answers
356 views

How to include material flow in job shop scheduling problem (as constraint?)?

I am reading about https://en.wikipedia.org/wiki/Job-shop_scheduling and https://en.wikipedia.org/wiki/Optimal_job_scheduling. Such scheduling is possible if we know the number (amount) of operations (...
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  • 119
2 votes
0 answers
26 views

Lexicographic flow property and consequences

I am struggling with the understanding of the (quite famous) paper on lexicographical flows: Maximal, Lexicographic, and Dynamic Network Flows - by Edward Minieka ...
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4 votes
1 answer
321 views

pyomo/ipopt: nonlinear network optimization not converging

The question (very short version) Why can I not decrease the lower boundary for the decision variable model.v_dot (see implementation) below 30 ? As soon as I do so,...
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2 votes
1 answer
106 views

Advanced Transportation optimization problem

I'm trying to explore OR problems and wanted to know what algorithm fits here. Consider fuel stations with lithium ion battery that service replacing battery at each refuel ( you replace your batters ...
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3 votes
0 answers
175 views

Multi-commodity (Integral) flow problem with unit demand; complexity

It seems to me that a multi-commodity flow problem should be treated differently when the commodity demand constraint is an equivalence and not a lower-bound. Consider a unit demand. Apparently, in ...
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3 votes
1 answer
99 views

Extreme points for capacitated flow polytope

Given a network as a directed acyclic graph with a source and sink, and non-negative edge capacities, I am interested in the extreme source-sink flows. More formally, I am interested in a ...
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  • 1,311
4 votes
1 answer
170 views

Problems modeling a constraint in network design problem

I'm working on a network design problem where the objective is to minimise the network design cost. Given a graph G = (V, E) and a set of point-to-point demands K, the task is to route the demands and ...
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  • 143
4 votes
1 answer
144 views

Max-flow of undirected graph

I know that a max flow problem for undirected graphs has a standard reduction to directed graphs in order to apply Ford–Fulkerson algorithm and getting a feasible solution. Here's my doubt. An ...
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4 votes
1 answer
365 views

Best way to add dummy to transportation problem? Zero cost will be always chosen first?

I know that an unbalanced transportation problem could be made a balanced transportation problem by adding a dummy node which equals the difference between demand and supply. In literature, dummy ...
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5 votes
1 answer
203 views

How to model this multi-commodity flow problem with a twist?

Problem description I am working on a variation of the multicommodity flow problem on a graph $G=(V,E)$. I have requests $r_{uv}$ for pairs of nodes $(u,v), u\in V, v\in V$, which means that $r_{uv}$ ...
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5 votes
4 answers
595 views

How to visit a subset of network nodes in a single trip?

I have a connected network where I want to visit a set of destinations which may require visiting intermediate nodes as well because there may be no direct edge between source and destination nodes. I ...
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  • 61
2 votes
1 answer
106 views

Constraining flow into a node and out of a node using Min Cost Flow input

Let's say I have a graph $G=(V,E)$ where each vertex has an edge going both into and out of it (i.e. cyclic, akin to the kidney donor pair problem). Now suppose we have a capacity $u_{ij} = 1 \forall(...
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3 votes
1 answer
327 views

Min Cost Flow with lower bound reduction to MCF algorithm

We define the Min Cost Flow Problem with Lower Bounds (MCFPLB) as a generalization of the usual Min Cost Flow. The input consists of: a directed graph $G=(V,E)$ capacities $u_{ij} \geq 0$ for each ...
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1 vote
1 answer
155 views

Solving general minimum cost flow problems using only one demand and one supply node

This is a practice in using reduction. Suppose I have a solver that only allows input to a MCF that specifies only one demand and one supply node. How could I use this solver to solve general MCF ...
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1 vote
1 answer
601 views

Python Gurobipy Maximum Flow: How to model "source" and "sink" in constraints

I am attempting to capture the nature of a "source" node as well as a "sink" node where the net flow for each node aside from the source and sink is zero. Here is my code so far: <...
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0 votes
1 answer
85 views

Adding slack nodes to min cost network flows

I have the following question. I want to clarify couple of points. As you can see, total demand and total supply does not match, we do not have enough demand. What I want to ask is: Do we need to ...
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2 votes
0 answers
138 views

Solving transportation problem by the Network Simplex

I am trying to solve the following problem using Network Simplex method. But I have questions. My attempt: Basis Matrix$(B)$ Rows: 1, 2, 3, 4, 5 Column: (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) $$ \...
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2 votes
1 answer
78 views

Maximum flow minimum cut

For the following problem I am trying to find maximum flow and minimum cut: I found the maximum flow as 6 like this: $1-3-7-8:2 flows$ $1-2-5-8: 2 flows$ $1-3-4-5-8:1 flow$ $1-3-4-6-5-8:1 flow$ But I ...
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0 votes
1 answer
167 views

Minimum cost flow problem with negative cost arcs

As far as I know, if there is a directed arc with a negative cost, we change its direction to its opposite and get a positive cost. But in the following question, if we change the direction of the arc ...
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6 votes
1 answer
283 views

Extreme point and extreme ray of a network flow problem

"It is a well-known result in network flow theory that an extreme point and an extreme ray of the polyhedron defined by the convex hull of feasible region corresponds to a path and cycle (resp.) ...
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4 votes
1 answer
147 views

splittable multicommodity problem with CPLEX

I am working on a splittable multicommodity flow problem, where each commodity can be split into $k$ orders. However, I don't know the value of $k$ in advance. Here is my code. ...
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0 votes
1 answer
721 views

How to describe the traveling salesman problem with an integer programming model?

I'm trying to describe the travelling salesman problem as an integer programming model. I'm interested in the asymmetric version of the problem. The problem can be summarized as finding the optimal ...
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3 votes
1 answer
110 views

What is the difference between min- cut formulation and (bi) partitioning formulation?

I have a min-cut formulation and a bi-partitioning problem. The two problems focus on finding the minimal cut value separating the two partitions? So what are really the differences between the ...
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  • 413
4 votes
2 answers
215 views

How to solve this convex problem heuristically?

I have the following problem $$\max_{X_{i,j},i\in N_{U},j\in N_{B}}\sum_{i=1}^{N_U}\sum_{j=1}^{N_B}R_{i,j}X_{i,j}$$ $$\text{subject to}$$ $$a_{\min}\le\sum_{j=1}^{N_B}X_{i,j}\le a_{\max}, \forall i$$ $...
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3 votes
0 answers
90 views

Simplex method on graphs: How do I find a basic solution using the Ford-Fulkerson algorithm?

I'm tasked with solving a minimal cost flow problem. I'm asked to first use the Ford-Fulkerson algorithm on my graph to find a basic solution that will then be used to do the simplex method on that ...
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0 votes
1 answer
206 views

How can I find the shortest path for all nodes in a graph from a source $s$?

This is the shortest path problem. I've used a model where we can find the shortest path between the source and a specified destination. The idea behind this model is that we assign a flow of 1 for ...
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