transmcp.gms : Transportation Model as Equilibrium Problem
Dantzig's original transportation model (TRNSPORT) is
reformulated as a linear complementarity problem. We first
solve the model with fixed demand and supply quantities, and
then we incorporate price-responsiveness on both sides of the
market.
Reference:
- Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, 1963.
Small Model of Type: MCP
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$Title Transportation model as equilibrium problem (TRANSMCP,SEQ=126)
$Ontext
Dantzig's original transportation model (TRNSPORT) is
reformulated as a linear complementarity problem. We first
solve the model with fixed demand and supply quantities, and
then we incorporate price-responsiveness on both sides of the
market.
Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions.
Princeton University Press, Princeton, New Jersey, 1963.
$Offtext
Sets
i canning plants / seattle, san-diego /
j markets / new-york, chicago, topeka / ;
Parameters
a(i) capacity of plant i in cases (when prices are unity)
/ seattle 350
san-diego 600 /,
b(j) demand at market j in cases (when prices equal unity)
/ new-york 325
chicago 300
topeka 275 /,
esub(j) price elasticity of demand (at prices equal to unity)
/ new-york 1.5
chicago 1.2
topeka 2.0 /;
Table d(i,j) distance in thousands of miles
new-york chicago topeka
seattle 2.5 1.7 1.8
san-diego 2.5 1.8 1.4 ;
Scalar f freight in dollars per case per thousand miles /90/ ;
Parameter c(i,j) transport cost in thousands of dollars per case ;
c(i,j) = f * d(i,j) / 1000 ;
Parameter pbar(j) reference price at demand node j;
Positive variables
w(i) shadow price at supply node i,
p(j) shadow price at demand node j,
x(i,j) shipment quantities in cases;
Equations
supply(i) supply limit at plant i,
fxdemand(j) fixed demand at market j,
prdemand(j) price-responsive demand at market j,
profit(i,j) zero profit conditions;
profit(i,j).. w(i) + c(i,j) =g= p(j);
supply(i).. a(i) =g= sum(j, x(i,j));
fxdemand(j).. sum(i, x(i,j)) =g= b(j);
prdemand(j).. sum(i, x(i,j)) =g= b(j) * (pbar(j)/p(j))**esub(j);
* declare models including specification of equation-variable
* association:
Model fixedqty / profit.x, supply.w, fxdemand.p/ ;
Model equilqty / profit.x, supply.w, prdemand.p/ ;
* initial estimate:
p.l(j) = 1;
w.l(i) = 1;
Parameter report(*,*,*) summary report;
Solve fixedqty using mcp;
report(i,j,"fixed") = x.l(i,j);
report("price",j,"fixed") = p.l(j);
report(i,"price","fixed") = w.l(i);
* calibrate the demand functions:
pbar(j) = p.l(j);
* replicate the fixed demand equilibrium using flexible demand func:
Solve equilqty using mcp;
report(i,j,"flex") = x.l(i,j);
report("price",j,"flex") = p.l(j);
report(i,"price","flex") = w.l(i);
* compute a counter-factual equilibrium using fixed demand func:
c("seattle","chicago") = 0.5 * c("seattle","chicago");
Solve fixedqty using mcp;
report(i,j,"fixed CF") = x.l(i,j);
report("price",j,"fixed CF") = p.l(j);
report(i,"price","fixed CF") = w.l(i);
* compute a counter-factual equilibrium using flexible demand func:
Solve equilqty using mcp;
report(i,j,"flex CF") = x.l(i,j);
report("price",j,"flex CF") = p.l(j);
report(i,"price","flex CF") = w.l(i);
Display report;
execute_unload 'mcpReport.gdx', report;
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