\$title Stochastic Benders - Sequential GAMS Loop (SPBENDERS1,SEQ=418) \$onText This example demonstrates a stochastic Benders implementation for the simple transport example. This is the first example of a sequence of stochastic Benders implementations using various methods to solve the master and subproblem. This first example implements the stochastic Benders algorithm using sequential solves of the master and subproblems in a GAMS loop. Keywords: linear programming, stochastic Benders algorithm, transportation problem \$offText Set i 'factories' / f1*f3 / j 'distribution centers' / d1*d5 /; Parameter capacity(i) 'unit capacity at factories' / f1 500, f2 450, f3 650 / demand(j) 'unit demand at distribution centers' / d1 160, d2 120, d3 270, d4 325, d5 700 / prodcost 'unit production cost' / 14 / price 'sales price' / 24 / wastecost 'cost of removal of overstocked products' / 4 /; Table transcost(i,j) 'unit transportation cost' d1 d2 d3 d4 d5 f1 2.49 5.21 3.76 4.85 2.07 f2 1.46 2.54 1.83 1.86 4.76 f3 3.26 3.08 2.60 3.76 4.45; \$ifThen not set useBig Set s 'scenarios' / lo, mid, hi /; Table ScenarioData(s,*) 'possible outcomes for demand plus probabilities' d1 d2 d3 d4 d5 prob lo 150 100 250 300 600 0.25 mid 160 120 270 325 700 0.50 hi 170 135 300 350 800 0.25; \$else \$ if not set nrScen \$set nrScen 10 Set s 'scenarios' / s1*s%nrScen% /; Parameter ScenarioData(s,*) 'possible outcomes for demand plus probabilities'; option seed = 1234; ScenarioData(s,'prob') = 1/card(s); ScenarioData(s,j) = demand(j)*uniform(0.6,1.4); \$endIf * Benders master problem \$if not set maxiter \$set maxiter 25 Set iter 'max Benders iterations' / 1*%maxiter% / itActive(iter) 'active Benders cuts'; Parameter cutconst(iter) 'constants in optimality cuts' / #iter 0 / cutcoeff(iter,j) 'coefficients in optimality cuts' / #iter.#j 0 /; Variable ship(i,j) 'shipments' product(i) 'production' received(j) 'quantity sent to market' zmaster 'objective variable of master problem' theta 'future profit'; Positive Variable ship; Equation masterobj 'master objective function' production(i) 'calculate production in each factory' receive(j) 'calculate quantity to be send to markets' optcut(iter) 'Benders optimality cuts'; masterobj.. zmaster =e= theta - sum((i,j), transcost(i,j)*ship(i,j)) - sum(i, prodcost*product(i)); receive(j).. received(j) =e= sum(i, ship(i,j)); production(i).. product(i) =e= sum(j, ship(i,j)); optcut(itActive).. theta =l= cutconst(itActive) + sum(j, cutcoeff(itActive,j)*received(j)); product.up(i) = capacity(i); Model masterproblem / all /; * Benders' subproblem Variable sales(j) 'sales (actually sold)' waste(j) 'overstocked products' zsub 'objective variable of sub problem'; Positive Variable sales, waste; Equation subobj 'subproblem objective function' selling(j) 'part of received is sold' market(j) 'upperbound on sales'; subobj.. zsub =e= sum(j, price*sales(j)) - sum(j, wastecost*waste(j)); selling(j).. sales(j) + waste(j) =e= received.l(j); market(j).. sales(j) =l= demand(j); Model subproblem / subobj, selling, market /; * Benders loop Scalar rgap 'relative gap' / 0 / lowerBound 'global lower bound' / -inf / upperBound 'global upper bound' / +inf / objMaster / 0 / objSub / 0 /; option limRow = 0, limCol = 0, solPrint = silent, solver = cplexd, solveLink = %solveLink.loadLibrary%; received.l(j) = 0; objMaster = 0; \$if not set rtol \$set rtol 0.001 loop(iter, objSub = 0; loop(s, demand(j) = scenarioData(s,j); solve subproblem maximizing zsub using lp; objSub = objSub + ScenarioData(s,'prob')*zsub.l; cutconst(iter) = cutconst(iter) + ScenarioData(s,'prob')*sum(j, market.m(j)*demand(j)); cutcoeff(iter,j) = cutcoeff(iter,j) + ScenarioData(s,'prob')*selling.m(j); ); itActive(iter) = yes; if(lowerBound < objMaster + objSub, lowerBound = objMaster + objSub;); rgap = (upperBound - lowerBound)/(1 + abs(upperBound)); break\$(rgap < %rtol%); solve masterproblem maximizing zmaster using lp; upperBound = zmaster.l; objMaster = zmaster.l - theta.l; ); abort\$(rgap >= %rtol%) 'need more iterations', lowerbound, upperbound; display 'optimal solution', lowerbound, upperbound;