Where every ton of board goes

Knock out one mill: extra network cost per year

Network cost is freight plus outside purchases valued at $150/t plus unmet demand at $400/t. Click a bar to see the re-routed map.

Markets that import sheets

One day of board, end to end (short tons; hover a band)

Loads on the move, per day

Conversions used

What a unilateral $70/t increase does to open-market volume

Integrated tonnage does not move: a company's own box plants keep buying from its own mills. The open-market tons are the ones that can walk, and in a market where the next mill is $5/t away, they all do. Real increases hold because the producers announce them together and the index follows.

Published price points

The loop in numbers

Named plants from the EPA Facility Registry

Proxy vs measured: demand share by state

Board demand by Census division and end use (Mt, allocated)

Model vs record

Error bars

Did freight position predict the 2025 closures?

The model

Each mill is a source with a capacity and a set of grades. Each demand point wants a fixed number of tons of liner and of medium. Moving a ton costs freight, which rises with distance and switches from truck to rail near 450 road miles. Minimize freight subject to meeting demand without exceeding any mill: a transportation problem, the special case of min-cost flow where every arc runs from a source to a sink. Two things about this industry break the plain version, and both are fixed by adding nodes rather than changing the algorithm.

First, the big companies are integrated: a ton of IP board goes to the nearest IP plant. Each integrated owner gets its own copy of every demand point holding its share of the local market (IP 31%, Smurfit Westrock 20%, PCA 15.5%, Georgia-Pacific 9.5%, Pratt 6%, Green Bay 3%, Cascades 2%; 13% independents), served only by its own mills; an arc from the open market into each copy at $150/t lets an owner buy outside when short. Second, mills make specific grades: splitting each mill into a liner node and a medium node behind one capacity arc keeps kraft mills out of the medium market without a second commodity. With those expansions it is one min-cost flow, solved as an LP by HiGHS.

Duals. The multiplier on a mill's capacity row is what one more ton of that mill saves the network (the scarcity layer). The multiplier on a demand row is the delivered cost of the marginal ton in that county (the delivered-cost layer and the calculator's freight). Prospects rank, for every market, the active mills by delivered freight; the gap between the first two is the switching threshold. Price shocks add $70/t to one owner's open-market arcs only. The fiber loop is a second transportation LP: counties supply recovered OCC (92% of consumption, plus 3% for imported goods), recycled mills demand 1.15 t per ton of recycled board they ship, export docks take the AF&PA/USITC export tonnage, and freight decides. The maintenance calendar is an assignment problem: for each mill and month the cost of an outage is estimated as capacity dual × tons lost, from twelve no-outage solves, then assigned with at most 10 mills per month and two per owner (a small MILP), and re-solved month by month to measure the true penalty. Knockouts remove one mill and re-solve. The backtest puts 2026 demand on the 2024 mill set and asks which mills the LP leaves idle.

Assumptions that matter

• Demand geography is population tilted by state manufacturing share, rolled up to 441 points; two thirds liner, one third medium; 34.5 Mt domestic + 4.6 Mt exports in 2024, 32.2 + 3.6 in 2026. Canadian mills sell 35% of capacity south; Mexican mills are listed, not modeled.
• Owner shares of the box market are national; a state-level plant count by owner (in the 10-K property lists) would make them regional.
• Mill capacities are public figures of mixed vintage; owner totals reconcile to the 10-Ks, single IP or PCA legacy mills could be off by 10%.
• Freight is two straight lines with a diesel factor. No lanes, backhaul, car supply, transload economics or contracts; no swaps beyond the $150/t outside-purchase arc.
• The outage calendar (9 days kraft, 5 recycled, 3% unplanned) and the monthly demand index are the author's; companies' actual outage months are in their earnings guidance.

The core of the LP, in Python

owners = [k for k in owner_shares if (M.owner==k).any()]
s_open = 1.0 - sum(owner_shares[k] for k in owners)
grades = [("L", grade_split[0], liner_ok), ("M", grade_split[1], medium_ok)]
# ---- variable bookkeeping
cols = []  # (kind, k, m, j, g) ; kind in x,y,v,z,u
c_obj = []
rows_cap, rows_own, rows_open, rows_exp = [], {}, {}, []
A_r, A_c, A_v = [], [], []
def addvar(kind, k, m, j, g, c):
    cols.append((kind, k, m, j, g)); c_obj.append(c); return len(cols)-1
# constraint indices
cap_row = {m: m for m in range(nM)}
nrow = nM
own_row = {}
for k in owners:
    for j in range(J):
        for gi,(g,_,_) in enumerate(grades):
            own_row[(k,j,g)] = nrow; nrow += 1
open_row = {}
for j in range(J):
    for g,_,_ in grades:
        open_row[(j,g)] = nrow; nrow += 1
exp_row = {p: nrow+p for p in range(len(PORTS))}; nrow += len(PORTS)
def coef(r, cidx, v): A_r.append(r); A_c.append(cidx); A_v.append(v)
owner_of = M.owner.values
for m in range(nM):
    if cap[m] <= 0: continue
    for g, phi, ok in grades:
        if not ok[m]: continue
        for j in range(J):
            # open-market arc
            ci = addvar("y", None, m, j, g, cost[m,j] + surcharge[m]); coef(cap_row[m], ci, 1); coef(open_row[(j,g)], ci, 1)
            k = owner_of[m]
            if k in owner_shares and k in owners:
                ci = addvar("x", k, m, j, g, cost[m,j]); coef(cap_row[m], ci, 1); coef(own_row[(k,j,g)], ci, 1)
    if kraft[m]:
        for p in range(len(PORTS)):
            ci = addvar("z", None, m, None, p, pcost[m,p]); coef(cap_row[m], ci, 1); coef(exp_row[p], ci, 1)
for k in owners:
    for j in range(J):
        for g,_,_ in grades:
            ci = addvar("v", k, None, j, g, premium); coef(own_row[(k,j,g)], ci, 1); coef(open_row[(j,g)], ci, -1)
for j in range(J):
    for g,_,_ in grades:
        ci = addvar("u", None, None, j, g, unmet_penalty); coef(open_row[(j,g)], ci, 1)
nvar = len(cols)
A = coo_matrix((A_v, (A_r, A_c)), shape=(nrow, nvar)).tocsr()
# split into inequality (capacity rows) and equality rows
A_ub = A[:nM]; b_ub = cap
A_eq = A[nM:]
b_eq = np.zeros(nrow-nM)
for k in owners:
    for j in range(J):
        for g, phi, _ in grades:
            b_eq[own_row[(k,j,g)]-nM] = owner_shares[k]*phi*D[j]
for j in range(J):
    for g, phi, _ in grades:
        b_eq[open_row[(j,g)]-nM] = s_open*phi*D[j]
for p in range(len(PORTS)):
    b_eq[exp_row[p]-nM] = export_tons*PORTS.share.values[p]
res = linprog(np.array(c_obj), A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, bounds=(0,None), method="highs")