Conditioning And Weight Training
Conditioning And Weight Training HPER E1210
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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 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 Can we diagram this process Purdue University Ag Econ 652 PCLP DSP cont d Purdue University Ag Econ 652 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 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 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 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 PCLP DSP cont d To track yield the plantharvest scheduling variable must be indexed by planting period harvesting period and state ACREPPerHPerState 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 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 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 PCLP DSP cont d The objective becomes expected net return OBJECTIVE NETREV E SumState1CardState SumPPerHPerYieIdPperHperState YiedPPerHPerStatePrice PHCostACREPPerHPerState Purdue University Ag Econ 652 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 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 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 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 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 PCLP DSP cont d Compare the solution forthe plowing schedule between the PCLP model and PCLP DSP PCLP 148 VARIABLE PLOWL MARAPR69936 APR3 5064 PCLP DSP 175 VARIABLE PLOWL MARAPR69936 APR3 5064 Purdue University Ag Econ 652 PCLP DSP cont d PlantHarvest Schedule 176 VARIABLE ACREL Acreage S1 S2 S3 S4 S5 APR3SEP4 7421 7421 7421 7421 7421 APR40CT1 26445 23507 21156 20474 16702 APR4 OCT2 14 32 APR4NOV1 549 4919 7269 7952 11723 MAY10CT2 28426 25398 21829 20793 15067 MAY1NOV1 3028 6596 763313358 MAY20CT2 10727 10727 10727 10727 10727 Purdue University Ag Econ 652 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 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 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 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 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 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 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 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 I13 Scheduling Example Farm Planning GAMS model similar to the PCLP model used for the Top Crop Farmer Workshop Goal schedule machinery and land resources Q Why is scheduling important I A Efficiency in resource use translates into higher profits for the producer Purdue University Ag Econ 652 Farm Planning Example cont d Model Uses Allows producer to develop a rough operational plan Allows user to ask what if questions about expanding the resource base rent or buy landmachines or to see how a new crop would fit into the current system Allows policy analysts to simulate producer responses to alternative policies Purdue University Ag Econ 652 Farm Planning Example cont d Critical features Corn is the only crop grown Crop yields are sensitive to planting and harvesting dates Adjustment to yield due to timing is captured through a parameter YIELDPPERHPER Machinery resources are scarce enough that several potential bottlenecks can occur Purdu 652 e niversltyAg con Farm Planning Example cont d This is scheduling over time periods are on the order of 23 weeks in length and ofvariable length This length of period problem speci c is selected compromise between Having the time periods so short that the randomness of weather will regularly override any plan or Having the time periods so long that resource bottlenecks cannot be identified Purdue University Ag Econ 652 Farm Planning Example cont d SETS PER Periods MARAPRAPR3APR4MAY1MAY2MAY3 MAY4MAYJJUN1JUN2JUN3 JUNJLYJULY JYAUGAUGSPSEP40CT10CT2NOV1NOV2 The periods are defined as follows MARAPR1Mar1Apr21 APR31Apr 2225 APR41Apr 26May 2 MAY11May 39 MAY21May1016 MAY31May1723 MAY41May 2430 MAYJ1May 31Jun 6 JUN11Jun 713 JUN22Jun1420 JUN32Jun 2127 JUNJLYzJun 28Jul 4 JULY1Ju 5Jul 11JYAUGJuI12 Au929 AUGSP1Aug 30Sep19 SEP4Sep 2026 OCT1Sep 27Oct10 OCT2Oot1131NOV1Nov121NOV2Nov 22Dec 5 Purdue University Ag Econ 652 Farm Planning Example cont d Only some of these periods are suitable for Planting Harvesting SETS PPERPER Planting periods I APR3APR4MAY1MAY2MAY3MAY4MAYJ HPERPER Harvesting periods I SEP40CT10CT2NOV1NOV2 Purdue University Ag Econ 652 Farm Planning Example cont d Hours available per period for field work are reflected in several parameters Tractor a scalar denoting the number of tractors HrsPerDayT a scalar denoting the number of hours the tractor can be run per day DaysPer a parameter denoting the number of days available for field work by period I ComthsHper a parameter denoting the number of hours the combine is available by penod Purdue University Ag Econ 652 Farm Planning Example cont d Three things must be scheduled in this example I Plowing the eld to prepare for planting Planting the corn which must occur after plowing Harvesting the corn Purdue University Ag Econ 652 Farm Planning Example cont d Our problem variables are VARIABLE NETREV Total net revenue POSITIVE VARIABLES The following variables are in units of acres PLOWPer Acres plowed in period per ACREPerPer Acreage on PperHper sched Why is PLOW indexed only by the period when plowing occurs but ACRE is indexed by both planting period and harvest period Purdue University Ag Econ 652 Farm Planning Example cont d PLOWPer It does not matter when the plowing is done as long as it is prior to planting ACREPperHper The schedule of planting and harvesting determines yield so it is not possible to independently schedule planting and harvesting without losing important information Purdue University Ag Econ 652 Farm Planning Example cont d I Model equations include the following I EQUATIONS OBJECTIVE Objective function net revenue LAND Land availability constraints TRACTRPer Tractor constraints in planting pers SEQPer Sequencing of plowing and planting HARVESTPer Combine constraints in harvest pers Purdue University Ag Econ 652 Farm Planning Example cont d OBJECTIVE NETREV E SumPPerHPerYieldPPerHPer YieldPPerHPerPrice PHCost ACREPPerHPer Notice that the logical value of YieldPperHper is used to suppress unuseful variables Price denotes the price of com PHCost denotes the variable costs of production Purdue University Ag Econ 652 Farm Planning Example cont d I LAND I SumPPerHPerYieldPperHperACREPPerHPer I L LandRhs This constraint simply restricts the use of land to be no greaterthan the available land Note this constraint is static it would not be appropriate if several crops could be grown on the same piece of land during the year Purdue University Ag Econ 652 Farm Planning Example cont d Usage ofthe tractor resources must be scheduled Available tractor time has been addressed TractorHrsPerDayTDaysPER Usage of tractors for different operations is re ected in three scalars PIwTrac hours per acre to plow PItTrac hours per acre to plant HavTrac hours per acre to harvest Purdue University Ag Econ 652 Farm Planning Example cont d I TRACTRPer I leTracPlowPer I SumHPerYieldPerHperPltTracACREPerHPer I SumPPerYieldPPerPerHavTracACREPPerPer I L TractorHrsPerDayTDaysPER This constraint restricts uses of tractor hours not to exceed available tractor time This constraint is dynamic with use and availability varying by period Purdue University Ag Econ 652 Farm Planning Example cont d I SEQPer I SumPeriodordPeriod le ordPerPLOWPeriod I G SumPPerordPPer le ordPer I SumHPeryieldPPerHPerACREPPerHPer This constraint enforces the sequencing of plowing and planting Note that Period and Per are aliases The lefthand side is an expression forthe number of acres plowed from the beginning ofthe planning horizon through period Per Purdue University Ag Econ 652 Farm Planning Example cont d I SEQPer I SumPeriodordPeriod le ordPerPLOWPeriod I G SumPPerordPPer le ordPer I SumHPeryieldPPerHPerACREPPerHPer The righthand side is an expression for the cumulative number of acres planted from the beginning ofthe planning horizon through period Per The cumulative nature of these constraints is typical of sequencing constraints Purdue University Ag Econ 652 Farm Planning Example cont d I HARVESTHPer I SumPPerYieldPperHper CombACREPPerHPer I L ComthsHPer This constraint limits the use ofthe combine resource in period HPerto no more than the available amount of combine time This equation is generated only ifyieldpperhper is strictly positive for some pper Purdue University Ag Econ 652 Farm Planning Example cont d Stochastic elements ignored Yieldprice variability Resource availability Bad weather can prevent field operations Machines may break Workers may not show up Purdue University Ag Econ 652 Farm Planning Example cont d The goal is pro t maximization Some activities must be sequenced Plowing precedes planting Planting precedes harvesting Limitations on resources may be Static Land Dynamic Tractor Harvester Purdue University Ag Econ 652 Other Dynamic Models Optimal Growth Investment is also a process that occurs over time Investment produces new capital stock New capital stock does not appear instantly Depreciation is the gradual ie overtime decay of the productive capacity of capital stock Purdue University Ag Econ 652 National Energy Planning Model ETAMacro Model Manne etamacgms in GAMS library Energy Technology Assessment an engineering process based model of energy technology Macro a simple macro model Used for forecasting energy suppliesdemands over time Purdue University Ag Econ 652 ETAMacro cont d Time horizon 19902030 5year periods I Economywide objective maximize the discounted utility of consumption over time Utility is logarithmic and a fixed discount rate is used I Consumption isthe residual after investment in energy technologies and energy expenses Purdue University Ag Econ 652 ETAMacro cont d Dynamic features Capacity expansion is irreversible Investment drives capacity expansion Substitution possibilities are limited for installed capacity but exible for capacity additions puttyclay Depreciation of installed capital Purdue University Ag Econ 652 ETAMacro cont d Discounted sum of utility of consumption T T Z tuxct 213 1nCt t1 t1 Simple macro relationship between consumption and production Ct 2 Yr It ECt Consumption GDP Investment Energy Costs Purdue University Ag Econ 652 ETAMacro cont d Vintaged production inputsoutputs puttyclay formulation All inputs to production capital labor electric energy and nonelectric energy are vintaged k kn km 139dep I In ltl 139dep er en er1139dep n H nt 1139dep Purdue University Ag Econ 652 ETAMacro cont d Output is also vintaged yr ynr Yr 1139dep The new vintage output is a function of new vintage inputs according to the following relationship ynt A kl kntalntlia p yen ent nntli gtp IMP Initial stocks are given for output and inputs 652 urdue University Ag Econ ETAMacro cont d ETA Leontief production is used to describe energy production possibilities in the full model Here exogenous price schedules are given with nonelectric energy prices growing faster than electric energy prices Purdue University Ag Econ 652