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For nonnegative integer supplies $s_i$ and demands $d_j$ with equal totals, the complete transportation problem has an integral optimal shipment matrix for every finite real cost array. A feasible matrix exists, and its nonnegative entries are bounded by their row supplies, so the transportation polytope is compact and a linear objective attains its minimum.

Its constraints are the vertex-edge incidence equations of a bipartite graph. Negating the demand rows gives a directed incidence matrix, which is a totally unimodular matrix. After deleting one redundant equality, any independent square basis has determinant $+1$ or $-1$. An integer right side therefore gives integer basic coordinates by Cramer's rule. The Fundamental theorem of linear programming supplies an optimal basic feasible solution, proving integrality.

There is also an algorithmic view. The northwest corner method starts with integer shipments; a cycle pivot adds and subtracts an integer minimum along its alternating cycle, preserving integrality. Zero allocations may be needed to complete a degenerate basis. With forbidden routes, equality of total supply and demand is no longer enough for feasibility, though an existing transportation feasible polytope retains the same incidence-matrix integrality property.

Use vertices $1,2,3,4,5$. Reading each symmetric off-diagonal pair once gives the edge setHere $ij$ abbreviates the undirected edge $\{i,j\}$. The diagram displays these seven edges and no loops. The displayed square matrix is an adjacency matrix in the usual terminology: its two indices both label vertices. A vertex-edge incidence matrix would instead have one index for vertices and one for edges. This explains how the source's node-node terminology is being read.

The directed vertex-edge incidence matrix has one $+1$ and one $-1$ per nonloop column and a zero column for a loop. Every square subdeterminant belongs to $\{0,1,-1\}$. Induct on its size: a column with no nonzero entries makes the determinant zero; a column with one nonzero entry permits a cofactor expansion to a smaller determinant; and if every column has two nonzero entries, their opposite signs make the column sums zero, giving linearly dependent rows. Negating rows preserves absolute subdeterminants. Consequently the unsigned vertex-edge incidence matrix of a bipartite graph is totally unimodular: negate all rows in one part. The assignment problem is an instance, with row vertices and column vertices forming the two parts.