Align with Dantzig's (1963) original transport problem

[khaeru]

In both message_ix and ixmp, we use a linear programming (LP) problem from Dantzig (1963) for a simple test case. In the course of working on #451 (comment), I found that our implementation is not faithful to the original.

Specifically:

• The original report is available at https://www.rand.org/pubs/reports/R366.html. See Chapter 3 "Formulating a Linear Programming Model", Section 3-3 "A Transportation Problem", pp.35–41 (PDF pages 50–56).
• The text reads, in part:

  Cases available: 350 at Cannery I, 650 at Cannery II = 1000 total available

  Cases demanded: 300 at Warehouse A, 300 at Warehouse B, 300 at Warehouse C = 900 total required

  The excess production (100 cases) should be stored without shipment.

• We have the following numbers which do not match:

  message_ix/message_ix/testing/__init__.py

  Lines 410 to 426 in 9fb0b97

  	par["demand"] = make_df(
  	"demand",
  	**common,
  	node=["new-york", "chicago", "topeka"],
  	level="consumption",
  	value=[325, 300, 275],
  	unit="case",
  	)
  	par["bound_activity_up"] = make_df(
  	"bound_activity_up",
  	**common,
  	node_loc=["seattle", "san-diego"],
  	mode="production",
  	technology="canning_plant",
  	value=[350, 600],
  	unit="case",
  	)

  
  and

  message_ix/message_ix/tests/test_feature_bound_activity_shares.py

  Lines 46 to 53 in 9fb0b97

  	["seattle", "production", 350.0, 350.0],
  	["seattle", "to_new-york", 50.0, 0.0],
  	["seattle", "to_chicago", 300.0, 300.0],
  	["seattle", "to_topeka", 0.0, 0.0],
  	["san-diego", "production", 600.0, 600.0],
  	["san-diego", "to_new-york", 275.0, 325.0],
  	["san-diego", "to_chicago", 0.0, 0.0],
  	["san-diego", "to_topeka", 275.0, 275.0],

One consequence of this—as described in the linked comment—is that the problem does not have a unique solution. In #451 (possibly other PRs that change the GAMS code) this causes difficulty because the solver chooses a different solution with the same objective function value.

IMO there is no reason we can't adjust make_dantzig() to (a) align with the original statement of the problem and (b) have a single, unique solution. However, to do this may also require adjustments to expectations wherever this is used across our test suite (many places). #451 will contain adjustments narrowly focused on that PR, but I open this issue towards a general fix.

[glatterf42]

Oh, thanks for pointing this out! I never bothered looking at the original formulation, assuming the one we currently have was faithful.