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# 529 Class Note for HPER E1210 at Purdue

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This 13 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 23 views.

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Date Created: 02/06/15

Another DSP Example I Farm Planning with Uncertain Resource Utilization Coefficients Revisit the GAMS model similarto PCLP from Lecture 16 Modi cation is to accommodate the dependence of the harvesting rate on the yield reflecting that the combine must go slower ifyields are high than othenNise Purdue University Ag Econ 652 Lecture 19 1 PCLP DSP Yields are now stochastic Constraint coef cient reflecting combining rate is now stochastic depends on yield Harvesting planconstraints will have to be exible to allow ef cient operations at different yield levels I Can we diagram this process Purdue University Ag Econ 652 Lecture 19 2 PCLP DSP cont d Plow Plant Yield I ll D Purdue University Ag Econ 652 Lecture 19 PCLP DSP cont d Note that decisions harvest scheduling are made after the nal random event yields Recall the decision variables from the PCLPtype model PlowPer Acresto plow in period Per AcrePPerHPer Acres to plant in period PPer and harvest in period HPer Purdue University Ag Econ 652 Lecture 19 PCLP DSP cont d The random event that we account for isthe realization of ideal yields Ideal yields are either 120 135150 155 or 190 buacre each with equal probability Yield adjustments remain the same on a percentage basis regardless of which ideal yield occurs This information becomes available after plowing and planting have been completed Purdue University Ag Econ 652 Lecture 19 5 PCLP DSP cont d Define an index set to indicate the state of nature I SET State 8185 And let the ideal yield parameter be I PARAMETER YLDSTATE ideal bu per acre S11208213583150S415585190 The mean of this distribution and the realization for 83 are the same as the base yield for PCLP model Purdue University Ag Econ 652 Lecture 19 6 PCLP DSP cont d The schedule for plowing must be made independent ofthe state of nature PIowPPer The schedule for planting is also made independent ofthe state of nature I However the yield adjustment is tied to the combination of planting and harvesting date and The base yield shifts with the state of nature Purdue University Ag Econ 652 Lecture 19 7 PCLP DSP cont d To track yield the plantharvest scheduling variable must be indexed by planting period harvesting period and state ACREPPerHPerState I But the number of acres planted in any specific planting period must be the same across all states reflecting that the state is unknown when the planting schedule is determined Purdue University Ag Econ 652 Lecture 19 PCLP DSP cont d To reflect invariance of the planting schedule by date EQUATIONS PLANTPPerState Planting is equal cross states PLANTPPerState SumHPerYieldPPerHPerState SumHPerYieldPPerHPerState ACREPPerHPerStateE SumHPerYieldPPerHPerState ACREPPerHPer S1 Purdue University Ag Econ 652 Lecture 19 9 PCLP DSP cont d We retain flexibility to shift the harvest schedule as the yield has been effectively realized by the time harvesting begins I Let us go back through the PCLP equations and see how they have changed to incorporate the conditional nature of the problem Purdue University Ag Econ 652 Lecture 19 10 PCLP DSP cont d The objective becomes expected net return OBJECTIVE NETREV E SumState1CardState SumPPerHPerYieIdPperHperState YiedPPerHPerStatePrice PHCostACREPPerHPerState Purdue University Ag Econ 652 Lecture 19 11 PCLP DSP cont d Because the planting schedule is the same in every state it is sufficient to constrain land for a single state eg S1 LAND SumPPerHPer SumStateYieIdPperHperState ACREPPerHPer39S139 L Land Rhs Purdue University Ag Econ 652 Lecture 19 12 PCLP DSP cont d The tractor use constraint spans the entire time horizon including harvest periods I TRACTRPerState leTracPlowPer SumHPerYieldPerHperState PltTracACREPerHPerState SumPPerYieldPPerPerState HavTracACREPPerPerState L TractorHRSPERDAYTDAYSPER Purdue University Ag Econ 652 Lecture 19 13 PCLP DSP cont d As with the Land constraint it is suf cient to impose the sequencing constraint for plowing and planting for only one state SEQPer SumPeriodordPeriod le ordPer PLOWPeriod G SumPperordPper le ordPer SumHperyieldPperHper S1 ACREPperHper39S139 Purdue University Ag Econ 652 Lecture 19 14 PCLP DSP cont d The big difference between this model and the PCLP model is that the combine working rate depends on yield Here the combine working rate is speci ed in hoursbu SCALAR COMB Combine working rate hrs per bu 00022222 To get total hours used we multiply hrsbu buacre acre to get hrs Purdue University Ag Econ 652 Lecture 19 15 PCLP DSP cont d The combine use constraint becomes HARVESTHPerState SumPPerYieldPPerHPerState SumPPerYieldPPerHPerState CombYieldPPerHPerState ACREPPerHPerState L ComthsHPer Notice that the coef cients on ACRE vary by State and that the constraint is indexed by State Purdue University Ag Econ 652 Lecture 19 16 PCLP DSP cont d Compare the solution forthe plowing schedule between the PCLP model and PCLP DSP PCLP 148 VARIABLE PLOWL MARAPR 69936 APR3 5064 PCLP DSP 175 VARIABLE PLOWL MARAPR 69936 APR3 5064 Purdue University Ag Econ 652 Lecture 19 17 PCLP DSP cont d PlantHarvest Schedule I 176 VARIABLE ACREL Acreage I S1 S2 S3 S4 S5 I APR3SEP4 7421 7421 7421 7421 7421 I APR40CT1 26445 23507 21156 20474 16702 I APR4 OCT2 14 32 I APR4NOV1 549 4919 7269 7952 11723 I MAY10CT2 28426 25398 21829 20793 15067 I MAY1NOV1 3028 6596 763313358 I MAY20CT2 10727 10727 10727 10727 10727 Purdue University Ag Econ 652 Lecture 19 18 PCLP DSP cont d Notice that if we sum across harvest dates the ACRE variables have the following subtotals by plant date l S1 S2 S3 S4 S5 l APR3 7421 7421 7421 7421 7421 i APR4 28426 28426 28426 28426 28426 l MAY1 28426 28426 28426 28426 28426 l MAY2 10727 10727 10727 10727 10727 Why are these all the same across states How do they compare to the PCLP results Same Purdue University Ag Econ 652 Lecture 19 19 PCLP DSP cont d Now what ifwe sum ACRE across planting dates l S1 S2 S3 S4 S5 l SEP4 7421 7421 7421 7421 7421 i OCT1 26445 23507 21156 20474 16702 l OCT2 40585 36125 32557 3152 25795 l NOV1 549 7946 13866 15584 25082 How do these compare to the PCLP results 7421 21156 31477 14946 Notice the differences in 0th and Nov1 Purdue University Ag Econ 652 Lecture 19 20 I10 PCLP DSP cont d Now consider the shadow pricespenalty costs Lagrange multipliers and their interpretation Forthe Land constraint the marginal value of an additional acre is 18065 forthe PCLP model 18251 forthe PCLP DSP model This is the value of having one more acre of land for the current year only ie the value ofa rented acre Purdue University Ag Econ 652 Lecture 19 21 PCLP DSP cont d Now consider the shadow prices on combine capacity HARVESTM OCT1 7297 OCT2 5288 for PCLP OCT1S1 1459 OCT2S1 1079 for PCLP DSP Q Why so much lower for PCLP DSP A This is the conditional value increasing the RHS ofthe constraint only increases it one fifth ofthe time Purdue University Ag Econ 652 Lecture 19 22 I11 PCLP DSP cont d Notice that I 1459X5 7295 and 1079X5 5395 I These are pretty close to the values of 7297 and 5288 that we observed forthe PCLP model In general ifwe want the rental value ofa conditional factor we should divide by the probability ofthe condition occurring Purdue University Ag Econ 652 Lecture 19 23 DSP Summary Discrete Stochastic Programming Allows us to incorporate uncertainties not only in our objective but also in our constraints Both RHS and coef cients Reactions to realizations of random variables can be incorporated through the introduction of conditional variables and constraints Purdue University Ag Econ 652 Lecture 19 24 I12 DSP Summary The main focus should be on the initial decisions and resource values the unconditional variables and constraints associated with the leading decision node I When determining what one should do at decision node subsequent to the rst one should really build a new DSP model that looks fonNard in time from that decision node Purdue University Ag Econ 652 Lecture 19 25 I13

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