qsambal.gms : Social Accounting Matrix Balancing Problem QCP

**Description**

This is a QCP version of the gamslib model SAMBAL. A Social Accounting Matrix captures all the circular flows in an economy and is called balanced if the row totals equal the column totals. A sample problem illustrates the use of Nonlinear Programming to balance such matrices.

**Reference**

- Zenios, S A, Drud, A S, and Mulvey, J, Balancing some large Social Accounting Matrices with Nonlinear Programming. Tech. rep., Department of Civil Engineering, Princeton University, 1986.

**Small Model of Type :** QCP

**Category :** GAMS Model library

**Main file :** qsambal.gms

```
$title Social Accounting Matrix Balancing Problem (QSAMBAL,SEQ=285)
$onText
This is a QCP version of the gamslib model SAMBAL.
A Social Accounting Matrix captures all the circular flows in an
economy and is called balanced if the row totals equal the column
totals. A sample problem illustrates the use of Nonlinear Programming
to balance such matrices.
Zenios, S A, Drud, A S, and Mulvey, J, Balancing some large Social
Accounting Matrices with Nonlinear Programming. Tech. rep.,
Department of Civil Engineering, Princeton University, 1986.
Keywords: quadratic constraint programming, social accounting matrix, statistics
$offtext
Set i 'accounts' / lab, h1, h2, p1, p2 /;
Alias (i,j);
Table xb(i,j) 'original estimate'
lab h1 h2 p1 p2
lab 15 3 130 80
h1 na
h2 na
p1 15 130 20
p2 25 40 55 ;
Parameter
tb(i) 'original totals' / lab 220, (h1,h2) na, p1 190, p2 105 /
tw(i) 'weights for totals'
xw(i,j) 'weights for cells';
tw(i) = 1;
tw(i)$(tb(i) = na) = 0;
xw(i,j) = 1$xb(i,j);
xw(i,j)$(xb(i,j) = na) = 0;
display tw, xw;
Variable
x(i,j) 'estimated cells'
t(i) 'estimated totals'
dev 'deviations';
Equation
rbal(i) 'row balance'
cbal(j) 'column balance'
devsqr 'definition of square deviations';
rbal(i).. t(i) =e= sum(j$xb(i,j), x(i,j));
cbal(j).. t(j) =e= sum(i$xb(i,j), x(i,j));
devsqr.. dev =e= sum((i,j)$xw(i,j), xw(i,j)*sqr(xb(i,j) - x(i,j))/xb(i,j))
+ sum(i$tw(i), tw(i)*sqr(tb(i) - t(i))/tb(i));
Model bal / all /;
x.l(i,j) = xb(i,j)$xw(i,j);
t.l(i) = tb(i)$tw(i);
solve bal using qcp minimizing dev;
Parameter rep 'balancing summary report';
rep(i,j,'original') = xb(i,j);
rep(i,j,'estimate') = x.l(i,j);
display rep;
```