Scheduling Example Farm Planning. Farm Planning Example (cont d.)

Transcription

1 Scheduling Example Farm Planning GAMS model similar o he PC/LP model used for he Top Crop Farmer Workshop Goal schedule machinery and land resources Q: Why is scheduling imporan? A: Efficiency in resource use ranslaes ino higher profis for he producer Lecure 16 1 Farm Planning Example (con d.) Model Uses Allows producer o develop a rough operaional plan Allows user o ask wha if quesions abou expanding he resource base (ren or buy land/machines), or o see how a new crop would fi ino he curren sysem Allows policy analyss o simulae producer responses o alernaive policies Lecure

2 Farm Planning Example (con d.) Criical feaures: Corn is he only crop grown Crop yields are sensiive o planing and harvesing daes Adjusmen o yield due o iming is capured hrough a parameer YIELD(PPER,HPER) Machinery resources are scarce enough ha several poenial bolenecks can occur Lecure 16 3 Farm Planning Example (con d.) This is scheduling over ime periods are on he order of 2-3 weeks in lengh and of variable lengh This lengh of period (problem specific) is seleced compromise beween Having he ime periods so shor ha he randomness of weaher will regularly override any plan, or Having he ime periods so long ha resource bolenecks canno be idenified Lecure

3 Farm Planning Example (con d.) SETS PER Periods /MARAPR,APR3,APR4,MAY1,MAY2,MAY3, MAY4,MAYJ,JUN1,JUN2,JUN3, JUNJLY,JULY, JYAUG,AUGSP,SEP4,OCT1,OCT2,NOV1,NOV2/ * The periods are defined as follows: * MARAPR:Mar 1-Apr 21; APR3:Apr 22-25; APR4:Apr 26-May 2 * MAY1:May 3-9; MAY2 :May 10-16; MAY3:May * MAY4:May 24-30; MAYJ:May 31-Jun 6; JUN1:Jun 7-13 * JUN2:Jun 14-20, JUN3:Jun 21-27; JUNJLY:Jun 28-Jul 4 * JULY:Jul 5-Jul 11;JYAUG:Jul12-Aug29; AUGSP:Aug 30-Sep19; * SEP4:Sep 20-26; OCT1:Sep 27-Oc 10 * OCT2:Oc11-31;NOV1:Nov1-21;NOV2:Nov 22-Dec 5 Lecure 16 5 Farm Planning Example (con d.) Only some of hese periods are suiable for Planing Harvesing SETS PPER(PER) Planing periods /APR3,APR4,MAY1,MAY2,MAY3,MAY4,MAYJ/ HPER(PER) Harvesing periods /SEP4,OCT1,OCT2,NOV1,NOV2/ ; Lecure

4 Farm Planning Example (con d.) Hours available per period for field work are refleced in several parameers: Tracor a scalar denoing he number of racors HrsPerDayT a scalar denoing he number of hours he racor can be run per day Days(Per) a parameer denoing he number of days available for field work by period CombRhs(Hper) a parameer denoing he number of hours he combine is available by period Lecure 16 7 Farm Planning Example (con d.) Three hings mus be scheduled in his example: Plowing he field o prepare for planing Planing he corn which mus occur afer plowing Harvesing he corn Lecure

5 Farm Planning Example (con d.) Our problem variables are: VARIABLE NETREV Toal ne revenue ; POSITIVE VARIABLES * The following variables are in unis of acres. PLOW(Per) Acres plowed in period per ACRE(Per,Per) Acreage on Pper-Hper sched. ; Why is PLOW indexed only by he period when plowing occurs, bu ACRE is indexed by boh planing period and harves period? Lecure 16 9 Farm Planning Example (con d.) PLOW(Per): I does no maer when he plowing is done as long as i is prior o planing ACRE(Pper,Hper): The schedule of planing and harvesing deermines yield so, i is no possible o independenly schedule planing and harvesing wihou losing imporan informaion Lecure

6 Farm Planning Example (con d.) Model equaions include he following: EQUATIONS OBJECTIVE Objecive funcion (ne revenue) LAND Land availabiliy consrains TRACTR(Per) Tracor consrains in planing pers SEQ(Per) Sequencing of plowing and planing HARVEST(Per) Combine consrains in harves pers Lecure Farm Planning Example (con d.) OBJECTIVE.. NETREV =E= Sum((PPer,HPer)$Yield(PPer,HPer), (Yield(PPer,HPer)*Price P_HCos) *ACRE(PPer,HPer)) ; Noice ha he logical value of Yield(Pper,Hper) is used o suppress un-useful variables Price denoes he price of corn P_HCos denoes he variable coss of producion Lecure

7 Farm Planning Example (con d.) LAND.. Sum((PPer,HPer)$Yield(Pper,Hper), ACRE(PPer,HPer)) =L= LandRhs; This consrain simply resrics he use of land o be no greaer han he available land Noe his consrain is saic i would no be appropriae if several crops could be grown on he same piece of land during he year Lecure Farm Planning Example (con d.) Usage of he racor resources mus be scheduled Available racor ime has been addressed (Tracor*HrsPerDayT*Days(PER)) Usage of racors for differen operaions is refleced in hree scalars: PlwTrac hours per acre o plow PlTrac hours per acre o plan HavTrac hours per acre o harves Lecure

8 Farm Planning Example (con d.) TRACTR(Per).. PlwTrac*Plow(Per) + Sum(HPer$Yield(Per,Hper),PlTrac*ACRE(Per,HPer)) + Sum(PPer$Yield(PPer,Per),HavTrac*ACRE(PPer,Per)) =L= Tracor*HrsPerDayT*Days(PER) ; This consrain resrics uses of racor hours no o exceed available racor ime This consrain is dynamic wih use and availabiliy varying by period Lecure Farm Planning Example (con d.) SEQ(Per).. Sum(Period$(ord(Period) le ord(per)),plow(period)) =G= Sum(PPer$(ord(PPer) le ord(per)), Sum(HPer$yield(PPer,HPer),ACRE(PPer,HPer))) ; This consrain enforces he sequencing of plowing and planing (Noe ha Period and Per are aliases) The lef-hand side is an expression for he number of acres plowed from he beginning of he planning horizon hrough period Per Lecure

9 Farm Planning Example (con d.) SEQ(Per).. Sum(Period$(ord(Period) le ord(per)),plow(period)) =G= Sum(PPer$(ord(PPer) le ord(per)), Sum(HPer$yield(PPer,HPer),ACRE(PPer,HPer))) ; The righ-hand side is an expression for he cumulaive number of acres planed from he beginning of he planning horizon hrough period Per The cumulaive naure of hese consrains is ypical of sequencing consrains Lecure Farm Planning Example (con d.) HARVEST(HPer).. Sum(PPer$Yield(Pper,Hper), Comb*ACRE(PPer,HPer)) =L= CombRhs(HPer) ; This consrain limis he use of he combine resource in period HPer o no more han he available amoun of combine ime This equaion is generaed only if yield(pper,hper) is sricly posiive for some pper Lecure

10 Farm Planning Example (con d.) Sochasic elemens ignored: Yield/price variabiliy Resource availabiliy Bad weaher can preven field operaions Machines may break Workers may no show up Lecure Farm Planning Example (con d.) The goal is profi maximizaion Some aciviies mus be sequenced Plowing precedes planing Planing precedes harvesing Limiaions on resources may be Saic (Land) Dynamic (Tracor, Harveser) Lecure

11 Oher Dynamic Models Opimal Growh Invesmen is also a process ha occurs over ime Invesmen produces new capial sock New capial sock does no appear insanly Depreciaion is he gradual (i.e., over ime) decay of he producive capaciy of capial sock Lecure Naional Energy Planning Model ETA-Macro Model (Manne eamac.gms in GAMS library) Energy Technology Assessmen (an engineering process based model of energy echnology) Macro (a simple macro model) Used for forecasing energy supplies/demands over ime Lecure

12 ETA-Macro (con d.) Time horizon (5-year periods) Economy-wide objecive -- maximize he discouned uiliy of consumpion over ime Uiliy is logarihmic and a fixed discoun rae is used Consumpion is he residual afer invesmen (in energy echnologies) and energy expenses Lecure ETA-Macro (con d.) Dynamic feaures Capaciy expansion is irreversible Invesmen drives capaciy expansion Subsiuion possibiliies are limied for insalled capaciy, bu flexible for capaciy addiions (puy/clay) Depreciaion of insalled capial Lecure

13 ETA-Macro (con d.) Discouned sum of uiliy of consumpion: T = 1 β u ( C ) = β ln( C ) T = 1 Simple macro relaionship beween consumpion and producion: C = Y I EC (Consumpion = GDP Invesmen Energy Coss) Lecure ETA-Macro (con d.) Vinaged producion inpus/oupus (puy/clay formulaion) All inpus o producion (capial, labor, elecric energy and non-elecric energy) are vinaged: k = kn + k -1 (1-dep) l = ln + l -1 (1-dep) e = en + e -1 (1-dep) n = nn + n -1 (1-dep) Lecure

14 ETA-Macro (con d.) Oupu is also vinaged y = yn + y -1 (1-dep) The new vinage oupu is a funcion of new vinage inpus according o he following relaionship: yn ρ ρ α 1 α β 1 β [ γ kl ( kn ln ) + γ en( en nn ) ] = A 1/ ρ (Iniial socks are given for oupu and inpus) Lecure ETA-Macro (con d.) ETA Leonief producion is used o describe energy producion possibiliies in he full model Here, exogenous price schedules are given wih non-elecric energy prices growing faser han elecric energy prices Lecure