clearlak.gms : Scenario Reduction: ClearLake exercise

**Description**

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.

**Reference**

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

**Small Model of Type :** LP

**Category :** GAMS Model library

**Main file :** clearlak.gms

$TITLE Scenario Reduction: ClearLake exercise (CLEARLAK,SEQ=249) $ontext 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. $offtext SET p Precipitation levels in each month /low, normal, high/ t Time periods /dec,jan,feb,mar,apr/ baset(t) / dec / w Weather conditions /wet, dry/; SET tw(t,w) relates months to weather conditions / (jan,feb).wet (mar,apr).dry /; SET n nodes / n1 * n121 /; ALIAS (n,parent,child); SET root(n) root node / n1 / tn(t,n) map nodes to time periods anc(child,parent) ancestor mapping np(n,p) maps nodes to precipitation level leaf(n); np(n,p)$[mod(ord(n)-2,card(p)) eq ord(p)-1] = yes; np(root,p) = no; * display np; scalar tmp1, tmp2; tmp1 = 0; loop {t, tmp2 = (power[card(p), ord(t)]) / (card(p)-1); tn(t,n)$[ord(n) ge tmp1 and ord(n) lt tmp2] = yes; tmp1 = tmp2; }; * display tn; anc(child,parent)$[floor((ord(child)+1)/card(p)) eq ord(parent)] = yes; * display anc; leaf(n)$[ord(n) gt (power[card(p), card(t)-1] - 1) / (card(p)-1)] = yes; * display leaf; TABLE delta(w,p) Changes in reservoir level for each season low normal high dry -50 100 250 wet 50 150 350 ; PARAMETER pr(p) Probability distribution / low 0.25, normal 0.50, high 0.25 /; PARAMETER nprob(n) probability of being at any node; nprob(root) = 1; loop {anc(child,parent), nprob(child) = sum {np(child,p), pr(p)} * nprob(parent); }; * display nprob; * ndelta required for current scenRed implementation PARAMETER ndelta(n) water delta at each node; ndelta(n) = sum {(tw(t,w), np(n,p))$[tn(t,n)], delta(w,p)}; * display ndelta; tmp1 = sum {leaf, nprob(leaf)}; abort$[abs(tmp1-1) gt 1e-8] "Error in tree: leaf probabilities do not sum to 1"; SCALAR floodCost 'K$/mm for amounts over flood level' / 10 /; SCALAR lowCost 'K$/mm for amounts 250mm below flood level' / 5 /; SCALAR l0 'initial water level' /100/; VARIABLE ec 'Expected value of cost'; POSITIVE VARIABLE l(t,n) 'level of water in dam, EOP' r(t,n) 'mm released normally' f(t,n) 'mm of floodwater released' z(t,n) 'mm of water imported'; r.up(t,n) = 200; * water level l is relative to 250mm below flood stage l.up(t,n) = 250; l.fx(baset,n) = l0; SET nn(n) nodes in reduced tree sanc(child,parent) ancestor mapping for reduced tree canc(child,parent) computed ancestor mapping for reduced tree; PARAMETER snprob(n) probabilities for reduced tree; EQUATIONS ecdef, ldef(t,n); ecdef.. ec =e= sum {tn(t,nn), snprob(nn) * [floodCost * f(t,nn) + lowCost * z(t,nn)]}; ldef(tn(t,nn))$[not root(nn)].. l(t,nn) =e= sum {anc(nn,parent), l(t-1,parent)} + ndelta(nn) + z(t,nn) - r(t,nn) - f(t,nn); model mincost / ecdef, ldef /; $if set noscenred $goto noscenreduction * now let's shrink the node set $libinclude scenred.gms ScenRedParms('num_leaves') = sum {leaf, 1}; ScenRedParms('num_random') = 1; ScenRedParms('num_nodes') = card(n); ScenRedParms('num_time_steps') = card(t); * typically, one of the following two parameters is set ScenRedParms('red_percentage') = 0.5; * optional SCENRED input parameters: defaults are commented * ScenRedParms('num_stages') = ScenRedParms('num_time_steps'); * ScenRedParms('reduction_method') = 0; * ScenRedParms('where_random') = 10; * ScenRedParms('report_level') = 0; ScenRedParms('run_time_limit') = 60; execute_unload 'lakein.gdx', ScenRedParms, n, anc, nprob, ndelta; execute 'rm -f lakeout.gdx'; file opts / 'scenred.opt' /; putclose opts 'log_file = lakelog.txt' / 'input_gdx lakein.gdx' / 'output_gdx = lakeout.gdx'; execute 'scenred scenred.opt %system.redirlog%'; execute_load 'lakeout.gdx', ScenRedReport, snprob=red_prob, sanc=red_ancestor; display ScenRedReport; display snprob; nn(n) = snprob(n); display nn; * -- BEGIN consistency check of output -- * canc(anc(child,parent))$[nn(child)] = YES; display canc, sanc; SET chk(child,parent); chk(child,parent) = NO; chk(canc(child,parent)) = YES; chk(sanc(child,parent)) = NO; abort$[sum{chk, 1}] "Error in reduced tree: inconsistent output", chk; chk(sanc(child,parent)) = YES; chk(canc(child,parent)) = NO; abort$[sum{chk, 1}] "Error in reduced tree: inconsistent output", chk; tmp1 = sum {leaf(nn), snprob(leaf)} - 1; abort$[abs(tmp1) gt 1e-8] "Error in tree: leaf probabilities do not sum to 1", tmp1; * -- END consistency check of output -- * $goto donered $label noscenreduction * if no reduction done, assign entire tree to subsets nn(n) = yes; snprob(nn) = nprob(nn); $label donered solve mincost using lp min ec;