I get this question a lot. The smallest example is model number 1 from the GAMS model library [1]: a tiny transportation model (originally from [2]). The linear programming (LP) model looks like:
| \[\begin{align} \min \>&z \\ &z = \sum_{i,j} c_{i,j} x_{i,j}&&\text{(cost)}\\ &\sum_j x_{i,j} \le a_i \>\forall i &&\text{(supply)}\\ &\sum_i x_{i,j} \ge b_j \>\forall j &&\text{(demand)}\\ &x_{i,j} \ge 0 \end{align}\] |
The model has only two source nodes and three destination nodes and the solution looks like:
LOWER LEVEL UPPER MARGINAL ---- EQU cost . . . 1.0000 cost define objective function ---- EQU supply observe supply limit at plant i LOWER LEVEL UPPER MARGINAL seattle -INF 350.0000 350.0000 EPS ---- EQU demand satisfy demand at market j LOWER LEVEL UPPER MARGINAL new-york 325.0000 325.0000 +INF 0.2250 ---- VAR x shipment quantities in cases LOWER LEVEL UPPER MARGINAL seattle .new-york . 50.0000 +INF . LOWER LEVEL UPPER MARGINAL ---- VAR z -INF 153.6750 +INF . z total transportation costs in thousands of dollars |
There are quite a few things to say about this solution listing.
There are 6 equations in this model and 7 variables. The objective function is modeled as an equality here as GAMS does not really have the notion of an objective function: it has an objective variable. The objective variable \(z\) is a free variable i.e. it has bounds \(-\infty\) and \(\infty\). This is also a GAMS convention.
The dots indicate a zero value. In the cost equation we can see the lower- and upper-bound are the same (both zero) indicating an equality. Note that this equation is rewritten as \(z - \sum_{i,j} c_{i,j} x_{i,j} = 0\) hence the bounds having a zero value.
The column marked MARGINAL are duals (for the rows) and reduced costs for the variables.
The row SUPPLY(‘SEATTLE’) has a marginal with a value of EPS. This means: this row is non-basic but with a zero dual. The EPS is used to signal this is a non-basic row: it the value was zero we could have deduced the row is basic. All basic rows and variables have a marginal that is zero. To verify we can count the basic and non-basic variables:
ROW/COLUMN MARGINAL BASIS-STATUS cost 1.000 NB |
All the variables and rows that have a marginal of zero are basic. There are 6 of them. This is identical to the number of equations. This conforms to what we would expect. The other 7 rows and columns have a non-zero marginal and are non-basic.
The occurrence of an EPS in the marginals indicates dual degeneracy. This indicates are multiple optimal solutions (to be precise: multiple optimal bases). Indeed here is an alternative solution found with a different LP solver:
LOWER LEVEL UPPER MARGINAL ---- EQU cost . . . 1.0000 cost define objective function ---- EQU supply observe supply limit at plant i LOWER LEVEL UPPER MARGINAL seattle -INF 300.0000 350.0000 . ---- EQU demand satisfy demand at market j LOWER LEVEL UPPER MARGINAL new-york 325.0000 325.0000 +INF 0.2250 ---- VAR x shipment quantities in cases LOWER LEVEL UPPER MARGINAL seattle .new-york . . +INF EPS LOWER LEVEL UPPER MARGINAL ---- VAR z -INF 153.6750 +INF . z total transportation costs in thousands of dollars |
The term primal degeneracy is used to indicate that a basic row or column is at its bound (usually they are in between bounds). We see an example of that in row SUPPLY(‘SEATTLE’) here: this row is basic but the value 600 is equal to the upper bound. This solution is also dual degenerate: we have an EPS marginal for variable X(‘SEATTLE’,’NEW-YORK’).
Notes
Q: Can we enumerate all possible optimal solutions? Not so easy, but in [3] an interesting approach is demonstrated on this model.
Q: Can we make \(z\) a positive variable? Well, GAMS is not very consistent here.
You can not declare ‘positive variable z’ as GAMS will complain: Objective variable is not a free variable. But we can assign: ‘z.lo = 0;’. Don’t ask me why. In general I just keep the variable unbounded.
Q: Can we used ranged equations in GAMS? No.
The listing file seems to indicate we can set lower and upper-bounds on rows. But this is not really true. If you would assign values say ‘supply.up(i) = INF;’, GAMS will ignore this and reset the bounds on the equations when generating the model.
Q: What about NLPs? Roughly the same. There is one major deviation: NLP solutions typically contain superbasic variables. These are non-basic (i.e. with a nonzero marginal) but are between their bounds.
Q: What about MIPs? By default GAMS will do the following after a MIP terminates: fix all discrete variables to their current level and resolve as an LP. So you will see marginals from this final LP.
Q: Can a free variable be non-basic? In almost all cases a free variable in an LP solution will be basic (a free basic variable will in general never leave the basis). However, it is possible for such a variable to be non-basic in which case it is usually zero.
References
- https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_trnsport.html
- Dantzig, G. B., Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, 1963.
- http://yetanothermathprogrammingconsultant.blogspot.com/2016/01/finding-all-optimal-lp-solutions.html