| Primal formulation |
|---|
| \[\begin{align}\min& \sum_{i,j} \color{darkblue}{\mathit{cost}}_{i,j}\cdot\color{darkred}x_{i,j} \\ & \sum_j \color{darkred}x_{i,j}\le \color{darkblue}{\mathit{supply}}_{i}\perp \color{darkred}u_i \le 0 && \forall i \\ & \sum_i \color{darkred}x_{i,j}\ge \color{darkblue}{\mathit{demand}}_{j}\perp \color{darkred}v_j \ge 0 && \forall j \\ & \color{darkred}x_{i,j}\ge 0\end{align} \] |
Here \(\perp\) indicates "with dual ...". The duals for this model are \(\color{darkred}u_i\) and \(\color{darkred}v_j\). In the model, we have added the (optimal) signs of the duals. When we print the results of this model, we can see something like:
---- 56 VARIABLE x.L shipment quantities in cases
new-york chicago topeka
seattle 50.000300.000
san-diego 275.000275.000
---- 56 EQUATION supply.M observe supply limit at plant i
seattle EPS
---- 56 EQUATION demand.M satisfy demand at market j
new-york 0.225, chicago 0.153, topeka 0.126
Notes:
- The entries x.L are the optimal levels of x, and supply.M/demand.M are the optimal duals of the supply and demand equation.
- The duals of the supply equation are EPS for Seattle and 0 for San Diego. EPS means: this entry is numerical zero, but non-basic. I.e. this model is dual-degenerate. Indeed this model has two different optimal solutions.
- The duals have the correct sign.
| Dual formulation |
|---|
| \[\begin{align}\min& \sum_{i} \color{darkblue}{\mathit{supply}}_{i}\cdot\color{darkred}u_{i}+\sum_{j} \color{darkblue}{\mathit{demand}}_{j}\cdot\color{darkred}v_{j} \\ & \color{darkred}u_{i}+\color{darkred}v_{j}\le \color{darkblue}{\mathit{cost}}_{i,j}\perp \color{darkred}x_{i,j} \ge 0 && \forall i,j \\ & \color{darkred}u_{i}\le 0,\color{darkred}v_{j}\ge 0 \end{align} \] |
Note that \(\color{darkred}u_{i}\le 0\). Sometimes versions of this dual are given that have only non-negative variables. In that case \(\color{darkred}u_{i}\) is preceded by a \(-\) sign throughout the model.
--- 76 VARIABLE u.L dual of supply equation
( ALL 0.000 )
---- 76 VARIABLE v.L dual of demand equation
new-york 0.225, chicago 0.153, topeka 0.126
---- 76 EQUATION dualfeas.M
new-york chicago topeka
seattle 50.000300.000
san-diego 275.000275.000
Notes:
- The dual variables of this dual model correspond to the primal level of the primal model.
- The primal variables of the dual model are equal to the duals found in the solution of the primal model.
- The dual model is not dual-degenerate (no duals or reduced cost that are EPS; you need to inspect the complete solution for this).
| Equilibrium formulation |
|---|
| \[\begin{align}& \color{darkblue}{\mathit{supply}}_{i} \ge \sum_j \color{darkred}x_{i,j} \perp \color{darkred}u_{i}\ge 0 && \forall i \\ & \sum_i \color{darkred}x_{i,j} \ge \color{darkblue}{\mathit{demand}}_{j} \perp \color{darkred}v_{j}\ge 0 && \forall j \\ & \color{darkblue}{\mathit{cost}}_{i,j} + \color{darkred}u_{i} \ge \color{darkred}v_{j} \perp \color{darkred}x_{i,j}\ge 0 && \forall i,j\end{align} \] |
Note:
- This is a square system of (linear) equations plus complementarity conditions.
- There is no objective.
- Both the original primal variables \(\color{darkred}x_{i,j}\) and dual variables \(\color{darkred}u_i\) and \(\color{darkred}v_j\) are present as explicit variables in the model.
- We flipped the sign of \(\color{darkred}u_i\): all prices are positive. So the first constraint is rearranged.
- The first two constraints are market clearing equations. The last equation is a zero-profit condition.
---- 95 VARIABLE x.L shipment quantities in cases
new-york chicago topeka
seattle 50.000300.000
san-diego 275.000275.000
---- 95 VARIABLE u2.L dual of supply equation (sign flipped)
( ALL 0.000 )
---- 95 VARIABLE v.L dual of demand equation
new-york 0.225, chicago 0.153, topeka 0.126
References
- trnsport.gms : A Transportation Problem, https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_trnsport.html
- Karush–Kuhn–Tucker conditions, https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions
- transmcp.gms : Transportation Model as Equilibrium Problem, https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_transmcp.html
- Solving linear complementarity problems without an LCP solver, https://yetanothermathprogrammingconsultant.blogspot.com/2021/05/solving-linear-complementarity-problems.html
Appendix: GAMS model
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