$title Optimal Investment (LANDS,SEQ=188)

$onText
The following two-stage problem consists of determining the optimal
capacity investment in various types of power plants so as to meet
next period demands for electricity. Four power plants are considered
and they can operate in three different modes. The next period demand
for each of the three modes are to be met. There is a budget
constraint and also a constraint on the minimum total capacity.

Louveaux, F V, and Smeers, Y, Optimal Investments for Electricity
Generation: A Stochastic Model and a Test Problem. In
Ermoliev, Y, and Wets, R J, Eds, Numerical Techniques for
Stochastic Optimization Problems. Springer Verlag, 1988,
pp. 445-452.

This problem will be solved in two steps, we solve each scenario
separately and then all three scenarios together.

Keywords: linear programming, investment planning, stochastic programming,
          electricity generation
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Set
   i 'power plant type' / plant-1*plant-4 /
   j 'operating mode'   / mode-1*mode-3   /;

Parameter
   c(i) 'investment cost' / plant-1 10, plant-2 7, plant-3 16, plant-4 6 /
   d(j) 'energy demand'   / mode-1  na, mode-2 3,  mode-3 2              /;

Table f(i,j) 'operating cost'
             mode-1  mode-2  mode-3
   plant-1       40    24       4
   plant-2       45    27       4.5
   plant-3       32    19.2     3.2
   plant-4       55    33       5.5;

Scalar
   m 'min installed capacity' /  12 /
   b 'budget limit'           / 120 /;

Variable
   x(i)   'capacity installed'
   y(i,j) 'operating level'
   cost   'total cost';

Positive Variable x, y;

Equation
   defcost   'definition of total cost'
   mincap    'minimum installed capacity'
   bbal      'budget constraint'
   powbal(i) 'power balance'
   dembal(j) 'demand balance';

defcost..   cost =e= sum(i, c(i)*x(i)) + sum((i,j), f(i,j)*y(i,j));

mincap..    sum(i, x(i)) =g= m;

bbal..      sum(i, c(i)*x(I)) =l= b;

powbal(i).. sum(j, y(i,j)) =l= x(i);

dembal(j).. sum(i, y(i,j)) =g= d(j);

Model det / all /;

Set s 'nodes' / s-1*s-3 /;

Parameter
   dvar(s) / s-1 3,  s-2 5,  s-3 7  /
   prob(s) / s-1 .3, s-2 .4, s-3 .3 /
   repdet 'scenario report';

loop(s,
   d('mode-1') = dvar(s);
   solve det minimizing cost using lp;

   repdet('cost',s) = cost.l;
   repdet(i,s)      = x.l(i);
   repdet('prob',s) = prob(s);
   det.solPrint = %solPrint.quiet%;
);

* make model stochastic
Parameter ds(j,s) 'stochastic demand';

Positive Variable ys(i,j,s) 'operating level';

Equation
   defcosts     'definition of total cost'
   powbals(i,s) 'power balance'
   dembals(j,s) 'demand balance';

defcosts.. cost =e= sum(i, c(i)*x(i)) + sum((i,j,s), prob(s)*f(i,j)*ys(i,j,s));

powbals(i,s).. sum(j, ys(i,j,s)) =l= x(i);

dembals(j,s).. sum(i, ys(i,j,s)) =g= ds(j,s);

Model stoc / defcosts, mincap, bbal, powbals, dembals /;

ds(j,s)        = d(j);
ds('mode-1',s) = dvar(s);

solve stoc minimizing cost using lp;

repdet('cost','hedge') = cost.l;
repdet(i,'hedge')      = x.l(i);
display repdet;