$title Scenario Reduction: ClearLake exercise (CLEARLAKSP,SEQ=72)

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Exercise, p. 44:

The Clear Lake Dam controls the water level in Clear Lake, a
well-known resort in Dreamland.  The Dam Commission is trying to
decide how much water to release in each of the next four months.
The Lake is currently 150 mm below flood stage.  The dam is capable
of lowering the water level 200 mm each month, but additional
precipitation and evaporation affect the dam.  The weather near Clear
Lake is highly variable.  The Dam Commission has divided the months
into two two-month blocks of similar weather.  The months within each
block have the same probabilities for weather, which are assumed
independent of one another.  In each month of the first block, they
assign a probability of 1/2 to having a natural 100-mm increase in
water levels and probabilities of 1/4 to having a 50-mm decrease or a
250-mm increase in water levels.  All these figures correspond to
natural changes in water level without dam releases.  In each month
of the second block, they assign a probability of 1/2 to having a
natural 150-mm increase in water levels and probabilities of 1/4 to
having a 50-mm increase or a 350-mm increase in water levels.  If a
flood occurs, then damage is assessed at $10,000 per mm above flood
level.  A water level too low leads to costly importation of water.
These costs are $5000 per mm less than 250 mm below flood stage.  The
commission first considers an overall goal of minimizing expected
costs.  This model only considers this first objective.

Birge, R, and Louveaux, F V, Introduction to Stochastic Programming.
Springer, 1997.

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Sets p       Precipitation levels in each month / low, normal, high /
     t       Time periods / dec, jan, feb, mar, apr /;

Table deltastoch(t,p)  reservoir level change data for each season
             low     normal  high
     jan      50     150     350
     feb      50     150     350
     mar     -50     100     250
     apr     -50     100     250;

Parameter
     pr(p)     Probability distribution / low 0.25, normal 0.50, high 0.25 /
     floodCost 'K$/mm for amounts over flood level' / 10 /
     lowCost   'K$/mm for amounts 250mm below flood level' / 5 /
     linit     initial water level /100/
     delta(t)  random changes in reservoir level for each season;

Variable
     obj   cost
     l(t)  level of water in dam end of period
     r(t)  mm released normally
     f(t)  mm of floodwater released
     z(t)  mm of water imported;
Positive variable l, r, f, z;

r.up(t) = 200;
* water level l is relative to 250mm below flood stage
l.up(t) = 250;
l.fx('dec') = linit;
delta(t) = sum(p, pr(p)*deltastoch(t,p));

Equations defobj, ldef(t);

defobj.. obj =e= sum(t$(ord(t)>1), floodCost * f(t) + lowCost * z(t));

ldef(t)$(ord(t)>1)..
        l(t) =e= l(t-1) + delta(t) + z(t) - r(t) - f(t);

model mincost / defobj, ldef /;

file emp / '%emp.info%' /; emp.nd=2; put emp '* problem %gams.i%';
loop(t$(ord(t)>1),
   put / 'stage ' ord(t):2:0 ' ' l.tn(t) ' ' r.tn(t) ' ' f.tn(t) ' ' z.tn(t) ' ' ldef.tn(t) ' ' delta.tn(t)
   put / 'randvar ' delta.tn(t) ' discrete ';
   loop(p, put pr(p) deltastoch(t,p)));
putclose;

Set  s scenarios / s1*s81 /
Parameter
     s_delta(s,t) random variable realization
     s_l(s,t)     level of water in dam end of period by scenario
     s_r(s,t)     mm released normally by scenario
     s_f(s,t)     mm of floodwater released by scenario
     s_z(s,t)     mm of water imported by scenario
     s_obj(s)     cost by scenario;

Set  dict / s.    scenario.''
            delta.randvar. s_delta
            l.    level.   s_l
            r.    level.   s_r
            f.    level.   s_f
            z.    level.   s_z
            obj.  level.   s_obj /;

solve mincost min obj using emp scenario dict;