Choosing the mode of transport¶

Moving 180 tonnes of chemicals out of four depots, where a depot may reach a centre by rail or by road at different cost.

✔ Verified against the published optimum — 1715, from Guéret, Prins, Sevaux & Heipcke, Applications of Optimization with Xpress-MP §10.2.3.

The connection has a name, so two of them may join the same depot and centre and keep their own cost, minimum and capacity. That is the whole model. On p. 142 the book observes that its data "cannot be coded as a (two-dimensional) matrix: for instance the element COST\(_{ij}\) of a cost matrix can only define a single cost". It works around that with a fictitious node per mode per connection, six of them, turning each parallel pair into two paths through distinct intermediates.

Here the connection is the axis, so the six nodes are not needed. d2_c2_rail and d2_c2_road are two rows that disagree on cost (12 against 14) and on band (10–50 against unbounded). The book's graph carries 15 nodes. This model carries four.

The model¶

The tabs start from the instance's tables — one frame per parameter.

description: >-
  Moving 180 tonnes of chemicals out of four depots to three recycling centres,
  where a depot may reach a centre by rail or by road at different cost. The
  connection is a thing with a name, so two of them may join the same depot and
  centre and keep their own cost and their own band; a cost matrix indexed by
  depot and centre has one cell for the pair and cannot. The rail and road
  links between the same depot and centre disagree on all three numbers.
  Guéret, Prins, Sevaux and Heipcke, Applications of Optimization with
  Xpress-MP, section 10.2. Optimum 1715, published in section 10.2.3.

dimensions:
  depot:
    description: depots the chemicals leave from
    dtype: str
  connection:
    description: one way of reaching one centre from one depot, by rail or by road
    dtype: str

relations:
  origin:
    description: the depot a connection leaves
    key: connection
    values: depot

parameters:
  stock:
    description: tonnes standing at a depot
    dims: [depot]
  cost:
    description: cost per tonne moved over a connection
    dims: [connection]
  min_load:
    description: >-
      smallest delivery a connection accepts — rail carries between 10 and 50
      tonnes per delivery, and road is unconstrained
    dims: [connection]
  max_load:
    description: largest delivery a connection accepts
    dims: [connection]
  total_to_move:
    description: tonnes that have to leave the depots altogether
    dims: []

variables:
  moved:
    description: tonnes sent over a connection
    dims: [connection]
    bounds:
      lower: min_load
      upper: max_load

constraints:
  within_stock:
    dims: [depot]
    expression: sum(moved, by=origin, over=connection, into=depot) <= stock

  move_the_lot:
    dims: []
    expression: sum(moved, over=connection) == total_to_move

objective:
  sense: minimize
  description: total cost of the deliveries
  expression: sum(moved * cost)

The centre is in the connection's name and in nothing else. This problem sets no per-centre demand, only a total of 180 t, so no constraint reads where a connection ends. A destination coordinate would be a declaration nothing consumes, and the language warns about that, so it is not written. The pairing is still visible where it matters: two connections out of d2 differing only in mode, which is what a (depot, centre) table cannot hold.

The second leg was load-bearing in the book's formulation, because its fictitious nodes need conservation, which reads both ends of every arc. Reifying the connection removed that job. A problem with per-destination demand needs both legs. §12.3 of the same book is one, and is a port of its own.

The rail band is a hard bound here, as in the source. The prose says rail carries "at least 10 tonnes and at most 50 tonnes for any single delivery", which reads as semi-continuous: either nothing or at least ten. The book's own Mosel model writes flow(a) >= MINCAP(a) unconditionally, so every rail connection carries at least 10 t whether it is wanted or not. The published 1715 is that reading. The port matches the model that produced the number. The semi-continuous variant is #383.

What it exercises¶

sum(by=) through a relation whose target carries a constraint of its own, and per-label bounds read from data, where the label is a connection rather than a pair.