Management Science 461 Lecture 6 – Network Flow Problems October 21, 2008

Intro Go back to Red Dog Show how a number of DM models and techniques are special cases of Min-cost network flow problem

Example from MGTSC 352 Each arc has positive cost Supply nodes have negative values; demand nodes are positive Transshipment nodes have 0 demand, 0 supply Capacity of 10 per arc This is a balanced problem – why?

Model Define X ij = amount of product from node i to node j Objective Function: minimize total cost

Model For each node, we have a flow conservation constraint Incoming (Inflow) Supply Outgoing (Outflow) Demand In + Supply = Out + Demand In – Out = Demand – Supply In – Out ≥ Demand – Supply

All Constraints

New model Constraints hard to write out – but there’s an easier way. Node Arc Incidence Matrix cleans up this process but also reveals the structure of the problem Every row corresponds to an arc, and has at most one “1” and one “-1” entry Therefore, matrix is unimodular

Unimodularity References for all your unimodular needs:  Nemhauser & Wolsey, Integer & Combinatorial Optimization  Ahuja, Magnanti, and Orlin, Network Flows Unimodular matrix: “a nonsingular matrix with a determinate of ±1; a square matrix is totally unimodular if every nonsingular submatrix from it is unimodular.” So there.

Why do we care? Integer solutions without having to specify integer values Integer constraints force more costly Solver computation So important that other problems are modeled as transportation problems e.g. inventory

Summary Min cost network flow:  Objective function: min cost  Capacity on arcs  Flow conservation Special Cases:  Shortest path (source and sink nodes, no capacities)  Transshipment: No capacities  Transportation: All nodes either suppy or demand  Max flow problems – identify capacity constraints

Add location to the mix This is what you did in Red Dog, MGTSC 352 Minimize fixed cost of location + transportation cost Constraints as before (flow conservation and capacity) Add binary variables for facility location

Effects on flow conservation Add binary Yj as follows:  Sumproduct in objective function (fixed cost)  New flow balance constraint becomes IN – OUT ≥ DEMAND – SUPPLY*Y If Y = 0, make sure we satisfy demand (positive transportation costs ensure we never “pool” product at the node) If Y=1, we have the constraint as before, ensures we produce enough and ship it out.