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Multi-stage stochastic portfolio investment Description The model implements three alternatives to compute the EEV or
generalized quantities for a multi-stage problem. The recourse problem (RP) (file portfolio_deq.mos) and the deterministic problem formulation (file portfolio_det.mos) are implemented via submodels that are run iteratively from the main model in the file portfolio.mos. Further explanation of this example:
This model is discussed in Section 11.3.2.1 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.
portfolio.mos (!********************************************************************* Mosel Example Problems ====================== file portfolio.mos `````````````````` Multi-stage Stochastic Portfolio Investment Model (DEQ) with Dynamic VSS -- Computing the EEV for a multi-stage stochastic problem -- Example discussed in section 11.3.2.1 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, October 2020 based on the GAMS implementation in Portfolio.gms by J.Kallrath and T.Lohmann 2011 The model implements three alternatives to compute the EEV or generalized quantities for a multi-stage problem. It reproduces the results for EEV, EEV^, EDEV and VSSD derived in: "Case study. The investor’s problem" in Escudero et al. (2009, p.57) 1) EEV(t) problem: Fix previous stages, i.e. the money available = investment of previous period x yield of current period. In this approach all decision variables up to stage t-1 are fixed to the solution of the "Expected Value Problem". This can easily lead to infeasibilities. EEV1 = -1.51408 ( = RP ) EEV2 = -1.9631 EEV3 is infeasible EEV4 is infeasible 2) EEVhat(t) problem: In this approach we fix all variables to zero, which were zero in the "Expected Value Problem" up to stage t-1. Here we solve stochastic problems. EEV^1 = -1.51408 ( = RP ) EEV^2 = -1.9631 EEV^3 = -2.29698 EEV^4 = -3.787919375 3) EDEV(t) problem: The dynamic approach in which we solve many deterministic, very small problems. Here, we need to solve parts of the tree. EDEV1 = 4.74 EDEV2 = 1.40235 EDEV3 = -1.7235 EDEV4 = -3.78792 (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 'portfolio' uses "mmxprs", "mmjobs", "mmsystem" declarations I: set of string ! Investments T: set of string ! Stages TL: list of string ! (Ordered) list of stages S: range ! Scenarios H,HC: array(S,S,T) of boolean ! scenario-scenario-stage mapping P: array(S) of real ! Scenario probability RET: dynamic array(S,I,T) of real ! Return of investment i at begin of stage t in scenario s RExpanded: array(S,I,T) of real BDGT: real ! Budget CAP: real ! Capital target Solx: dynamic array(S,I,T) of real ! Solution values for variables x RP: real ! Objective value of recourse problem EV,EEVT: real ! Objective value of expected value problem WS: real ! Wait-and-see solution value EEV: array(T) of real ! EEV in stage t EEVhat: array(T) of real ! EEV^ in stage t EDEV: array(T) of real ! EDEV value of the dynamic EEV in stage t DEQmodel, DETmodel: Model ! Submodels ev: Event ! Event messages from submodels end-declarations !**** Submodel run with some error handling function solvesubprob(m: Model, mname: text, params: text):boolean returned:=true writeln("Solving ", mname, ", ", params) run(m, params) wait ev:=getnextevent if ev.class<>EVENT_END then writeln("Problem during execution of ", mname) exit(2) elif getexitcode(m) <> 0 then writeln(" No solution for ", mname) returned:=false else writeln(" ", mname, " solved") end-if end-function !**** Solve the recourse problem (RP), if successful retrieve data + result **** if compile("g","portfolio_deq.mos") <> 0 then writeln("Compilation of DEQ submodel failed.") exit(1) end-if load(DEQmodel, "portfolio_deq.bim") ! Run the submodel and retrieve results if the returned status is okay if not solvesubprob(DEQmodel, "DEQ submodel", "OUTFILE=shmem:res,DISPLAY=false") then exit(1) end-if initializations from "shmem:res" I T TL S RET RExpanded P BDGT CAP H HC RP as "Wealth" ! Objective function value of the recourse problem end-initializations fdelete("shmem:res") writeln("**** Recourse problem (RP) solution: ", RP) !******** Solve the Wait-and-see problem ******** declarations RWS: array(I,T) of real ! Expected value of investment i in stage t XWS: array(I,T) of real ! Amount invested into i in stage t WealthSol: array(S) of real ! Wealth obtained for scenario s end-declarations if compile("g","portfolio_det.mos") <> 0 then writeln("Compilation of DET submodel failed.") exit(1) end-if load(DETmodel, "portfolio_det.bim") forall(s in S) do forall(i in I, t in T) RWS(i,t) := sum(rs in S | HC(rs,s,t)) RET(rs,i,t) initializations to "shmem:ret" RWS as "RET" end-initializations ! Run the submodel and retrieve results if the returned status is okay if not solvesubprob(DETmodel, "DET submodel for scenario "+s, "OUTFILE=shmem:res,LOADR=true,VALR=shmem:ret,DISPLAY=false") then exit(0) end-if initializations from "shmem:res" XWS as "Solx" WS as "Wealth" end-initializations fdelete("shmem:res") fdelete("shmem:ret") writeln(" Solution: ", WS, " x=", XWS) WealthSol(s):=WS end-do ! Compute the Wait-and-See value WS := sum(s in S) P(s)*WealthSol(s) writeln("**** Wait-and-see problem (WS) solution: ", WS) !******** EV problem ******** declarations REV: array(I,T) of real ! Expected value of investment i in stage t XEV: array(I,T) of real ! Amount invested into i in stage t end-declarations forall(i in I, t in T) REV(i,t) := sum(s in S) P(s)*sum(rs in S | HC(rs,s,t)) RET(rs,i,t) initializations to "shmem:ret" REV as "RET" end-initializations ! Run the submodel and retrieve results if the returned status is okay if not solvesubprob(DETmodel, "DET submodel for EV problem", "OUTFILE=shmem:res,LOADR=true,VALR=shmem:ret,DISPLAY=false") then exit(1) end-if initializations from "shmem:res" XEV as "Solx" EV as "Wealth" end-initializations fdelete("shmem:res") fdelete("shmem:ret") writeln("**** EV problem solution: ", EV, " x=", XEV) !******** EEV(t) problem ******** forall(t in T) EEV(t):= -INFINITY; EEV('today'):= RP reset(Solx) forall(tt in 2..TL.size) do forall(s in S,i in I) Solx(s,i,TL(tt-1)):=XEV(i,TL(tt-1)) initializations to "shmem:solx" Solx end-initializations ! Run the submodel and retrieve results if the returned status is okay if solvesubprob(DEQmodel, "DEQ submodel for EEV("+tt+") problem ", "OUTFILE=shmem:res,LOADX=true,SOLX=shmem:solx,DISPLAY=false") then initializations from "shmem:res" EEVT as "Wealth" end-initializations EEV(TL(tt)):= EEVT writeln(" Solution: ", EEV(TL(tt))) end-if fdelete("shmem:res") fdelete("shmem:solx") if EEV(TL(tt))=-INFINITY then break; end-if end-do writeln("**** EEV(t) problem solutions: ", EEV) !******** EEVhat(t) problem ******** forall(t in T) EEVhat(t):= -INFINITY; EEVhat('today'):= RP reset(Solx) forall(tt in 2..TL.size) do forall(s in S,i in I | XEV(i,TL(tt-1))=0) Solx(s,i,TL(tt-1)):=0 initializations to "shmem:solx" Solx end-initializations ! Run the submodel and retrieve results if the returned status is okay if solvesubprob(DEQmodel, "DEQ submodel for EEVhat("+tt+") problem ", "OUTFILE=shmem:res,LOADX=true,SOLX=shmem:solx,DISPLAY=false") then initializations from "shmem:res" EEVT as "Wealth" end-initializations EEVhat(TL(tt)):= EEVT writeln(" Solution: ", EEVhat(TL(tt))) end-if fdelete("shmem:res") fdelete("shmem:solx") if EEVhat(TL(tt))=-INFINITY then break; end-if end-do writeln("**** EEVhat(t) problem solutions: ", EEVhat) !******** EDEV(t) problem: Forming groups out of representative scenarios ******** declarations XDyn: dynamic array(I,T) of real ! Solution values for variables x ActS: set of integer ! Active scenarios (corresponds to scenario groups) EndS: set of integer ! Leaves of the actual scenario group T1: array(T) of boolean ! Actual fixations T2: array(T) of boolean ! Free stages (initially all is free) PSum: real ! Sum of probabilities of scenarios in a scenario group Wealthst: array(S,T) of real ! Wealth (obj. value) of scenario s in stage t end-declarations ! Initialize flags forall(t in T) T1(t):=false forall(t in T) T2(t):=true reset(Solx) ! Loop over all stages forall(tt in TL) do T2(tt):=false ! Solve separate problems for all active (scenario) nodes EDEV(tt):= 0 ActS:= union(rs in S | or(s in S) HC(rs,s,tt)) {rs} forall(acts in ActS) do ! Fix x for the deterministic model with the data from the previous solve reset(XDyn) forall(i in I, t in T | T1(t)) XDyn(i,t):= Solx(acts,i,t) ! Determine all scenarios belonging to the current active scenario 'acts' EndS:= union(s in S | HC(acts,s,tt)) {s} ! Calculate the estimated values for the next stages and fill the previous stages PSum:= sum(s in EndS) P(s) forall(i in I,t in T | T2(t)) REV(i,t):= sum(rs in S,s in EndS | HC(rs,s,t)) P(s)/PSum*RET(rs,i,t) forall(i in I,t in T | T1(t)) REV(i,t):= RExpanded(acts,i,t) forall(i in I) REV(i,tt):= RET(acts,i,tt) initializations to "shmem:ret" REV as "RET" evaluation of array(i in I, t in T | exists(XDyn(i,t))) XDyn(i,t) as "Solx" end-initializations ! Execute the deterministic model with the new data if not solvesubprob(DETmodel, "DET submodel for DEV problem "+tt+"_"+acts, "OUTFILE=shmem:res,LOADR=true,VALR=shmem:ret,LOADX=true,SOLX=shmem:ret,DISPLAY=false") then exit(1) end-if ! Save the solution of the current run initializations from "shmem:res" XDyn as "Solx" EEVT as "Wealth" end-initializations fdelete("shmem:res") fdelete("shmem:ret") forall(s in EndS,i in I) Solx(s,i,tt):= XDyn(i,tt).sol EDEV(tt) += EEVT*PSum Wealthst(acts,tt):= EEVT writeln(" Solution: ", EEVT) end-do ! Update flag T1 (fix solution values for current stage) T1(tt):= true end-do writeln("**** EDEV problem solutions: ", EDEV) !******** Dynamic value of stochastic solution ******** declarations VSSD: array(T) of real ! Dynamic value of the stochastic solution for each time period t in the last stage end-declarations forall(t in T) VSSD(t):= maxlist(0, RP-EDEV(t)) ! VSSD has to be non-negative writeln("**** VSSD values: ", VSSD) end-model | |||||||||||||||

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