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Sludge production planning solved by recursion Description The two model versions show how to solve a production planning problem via iterative LP solves (sludge.mos) or by using a nonlinear formulation (sludge2.mos). Further explanation of this example:
This model is discussed in Section 11.2.1 of the book
'J. Kallrath: Business Optimization Using Mathematical Programming -
An Introduction with Case Studies and Solutions in Various Algebraic
Modeling Languages' (2nd edition, Springer, Cham, 2021, DOI 10.1007/978-3-030-73237-0).
Source Files By clicking on a file name, a preview is opened at the bottom of this page.
sludge.mos (!********************************************************************* Mosel Example Problems ====================== file sludge.mos ``````````````` Sludge problem illustrating recursion Example discussed in section 11.2.1 of J. Kallrath: Business Optimization Using Mathematical Programming - An Introduction with Case Studies and Solutions in Various Algebraic Modeling Languages. 2nd edition, Springer Nature, Cham, 2021 author: S. Heipcke, June 2018 (c) Copyright 2020 Fair Isaac Corporation Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. *********************************************************************!) model 'sludge' uses "mmxprs" declarations I= 2 ! Number of sources J= 2 ! Number of processing areas K= 3 ! Number of destinations (end points) C= 2 ! Number of components RI=1..I RJ=1..J RK=1..K RC=1..C LAM: array(RC,RJ) of real ! Assumed fraction of c at j (recursed) COST: array(RJ,RK) of real ! Per tonne disposal cost from j to k MAXD: array(RC,RK) of real ! Maximum tonnes disposal of c at k FRACT: array(RC,RI) of real ! Fraction of c in material from i AVAIL: array(RI) of real ! Availability at i end-declarations AVAIL::[100, 150] FRACT::[0.1, 0.2, 0.2, 0.15] COST::[2, 3, 1, 1, 5, 0.5] MAXD::[20.0, 20.0, 10.0, 15.0, 15.0, 15.0] LAM::[0.20, 0.15, ! Starting guesses for LAM(c,j) 0.12, 0.18] TOL:=0.001 setrandseed(7) declarations x: array(RI,RJ) of mpvar ! Amount (tonnes) sent from i to j y: array(RJ,RK) of mpvar ! Amount (tonnes) sent from j to k t: array(RJ) of mpvar ! Throughput at j (tonnes) q: array(RC,RJ) of mpvar ! Quantity of component c passing through j LastSol: real end-declarations ! Disposal cost Cost:=sum(j in RJ,k in RK) COST(j,k)*y(j,k) ! Get rid of all sludge at i forall(i in RI) Arid(i):= sum(j in RJ) x(i,j) = AVAIL(i) ! Mass balance into j forall(j in RJ) Tpx(j):= sum(i in RI) x(i,j) = t(j) ! Mass balance out of j forall(j in RJ) Tpy(j):= sum(k in RK) y(j,k) = t(j) ! Define quantities forall(c in RC,j in RJ) Q(c,j):= q(c,j) = sum(i in RI) FRACT(c,i)*x(i,j) writeln("***LAM initial values: ", LAM) LastSol:= -INFINITY; itercnt:=1 repeat ! Limits on disposal: not too much c at k forall(c in RC,k in RK) Mx(c,k):= sum(j in RJ) LAM(c,j)*y(j,k)<= MAXD(c,k) ! Solve the problem minimise(Cost) writeln("Iteration ", itercnt, ". Solution: ", getobjval) forall(i in RI,j in RJ) write(" x(", i, ",", j, "):", x(i,j).sol) writeln forall(j in RJ,k in RK) write(" y(", j, ",", k, "):", y(j,k).sol) writeln ! Calculate new LAM values or stop if converged forall(c in RC,j in RJ) do if t(j).sol<>0 then NEWL(c,j):=q(c,j).sol/t(j).sol else writeln("***Division by 0") break end-if end-do gap:=(sum(c in RC,j in RJ) abs(LAM(c,j)-NEWL(c,j))) / (sum(c in RC,j in RJ) LAM(c,j)) ifperturb:=LastSol>getobjval ! Perturbation to prevent cycling if gap<TOL then writeln("***Minimum deviation reached: ", gap, "%") break elif itercnt>=10000 then writeln("***Iteration limit reached. Difference: ", gap, "%") break else forall(c in RC,j in RJ) LAM(c,j):=NEWL(c,j)*if(ifperturb,0.9+0.2*random,1) writeln("***LAM updated values: ",LAM, " difference: ", gap, "%") end-if LastSol:=getobjval; itercnt+=1 until false end-model | |||||||||||||

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