page 1page 2page 3page 4page 5page 6page 7page 8page 9page 10page 11page 12page 13page 14page 15The Optimal Schedule forthe Opening of Buildings inan Office ComplexbyScot G. AndrusSenior Design inEngineering ManagementOREM4.390Presented29April,1982Assistant Professor:Dr. Richard S. Barrrrrrrrrorr1CONTENTSSubjectPageThe Problem2The Model2Definition of the Model3Definition of the Varibles3Assumptions4Definition of the Coefficients5The Formulated Model5The Optimal Solution6Conclusion7TablesI - The Optimal Rent-out Schedule9II- Net Profit per Building10III-Demand for Office Space11IV- Sale Value of the Buildings11V - Net Operating Income12VI- Construction Costs13TheProblemThis model was formulated to analyze a problempresented by a land development firm planning to constructa group of office buildings. The developer has acquiredapproximately fifty two acres of land in a prime northDallas location and is planning to construct seven officebuildings over the next several years. The goal of thismodel is to determine the schedule of building openingswhich will maximize the firm's profit.TheModelThe model deals with two distinct types of officebuildings, high rise and garden. The two types eachhave their own net incomes, demands for space, andconstruction costs.The model can, in fact, be consideredas two independent models.This fact is very helpfuland makes finding the optimal solution much easier, dueto the size of the final formulated model.In the formulation of the model, variables areseparated by building number, with buildings one, two,and three being high rise, and four, five, six, and sevenbeing garden type.Buildings one and three are essentialythe same,each having 350k sq. ft. of available space.Building two is slightly larger with 4.50k sq. ft.Buildings four and five are likewise essentially thesameeachhaving 60k sq. ft. Buildings six and seven are also2Definitionof theModelThis model as formulated is a mixed integer linearprogramming model with 105 variables and 85 constraints.The objective function finds.the optimal solution usingpresent value of the rent plus the present value of thesale value minus the present value of the construction costfor each of the seven buildings. The model is constrainedby the demand for each type of office space, by themaximum capacity of each building, and by the obviousrequirement that a building must be opened before it isrented out.DefinitionofVariablesThere are three sets of variables in this model,Ziy,Biy,and SALEi.Ziyinteger variable either 0 or 1. A.1 indicatesthat building i is to be opened in yeary.A 0 indicates no change in building's status.Biycontinuous variable indicating the number ofsquare feet to be rented in building i in yearySALEicontinuous variable indicating the sale value ofbuilding i in.year 7. It is found by multiplyingthe total number of sq. ft. rented out by theNOIi divided by the capitalization rate of 10%.Other constraining variables:CCiy- construction cost for building i in yearDly - demand for high rise space in yearD2y - demand for garden space in yeary.MCi - Maximum capacity of building i.Y•Y,3essentially the same with 75k sq. ft. each.The demand for each type of office space is basedon projections for Dallas as a whole, and this locationin particular.The actual figures were provided by thefirm involved.It should be noted the firm involved is primarilyinvolved in land development and is, therefore, notinterested in owning the buildings for an extended periodof time.They have specified that the buildings be soldby the model at the end of the time period, in this caseseven years.AssumptionsThere are several basic assumptions incorporatedinto the formulation of this model.The first assumption is that once a square foot ofspace is rented, it will remain rented for the remainderof the seven year time span. This is used so that incomecan be considered as a cash stream for n years, thuseliminating the need to keep track of rent for each year.This way it can be considered as a lump sum.The second assumption was one stipulated by the firminvolved; that is, there is to be no capital restrictionplaced on the opening of the buildings. They have enoughcapital to cover the construction of any or all of thebuildings at any time.4DefinitionofCoefficientsThe coefficient on the Ziyvariable is the-preseritvalue of the construction cost for buildingadjusted of inflation of 8% per year.The coefficient on the Biy variable is the presentvalue of the cash stream of rent for buildingiin yeary,adjusted for inflation of 8% per year.The coefficient on the SALEi variable is the presentvalue of the sale value of building i in year seven.The present value discount rate is 20% in all threecases.TheFormulatedModelThe formulated model can be mathematically representedin the following way:Maximize Profit=7r1Subject To:1)Z1Biy,<D1yBiy,<D2yForyfrom 1 to 72)E (Ziy*Diy),<.BiyFori from 1 to 77i 1 (PVy*Biy* OIiy+PV7*SAT,F;i-PVy*Ziy*cciy)7For i from 1 to 7(Ziy#Diy)`BlyBiy=MC iVor i from 1 to 7Ziy=1?nr i from 1 +n 7i in yeary,5(High rise space rentedless than or equal todemand)(Garden space rented lessthan or equal to demand)(The building must beopened before space isrented out)(Total space leased equalto building capacity)(The building is to be openedonce and only once)TheOptimalSolutionThe formulated model was solved using the LINDOsoftware package.LINDO was used because of its abilityto optimize the model using 0,1 variables.Due to the model's size and the increased complexityof solving an integer model, it was necessary to optimizethe model in two parts. The was acomplished by settingspecific Ziyvalues for i=1,2,3 and allowing the softwareto optimize for buildings four, five, six, and seven. Theoptimal values forZiyi=4-,5,6,7were then set, and thesoftware was allowed to optimize for buildings one, two,and three.The optimal value yields a present valuenet profit of$37,118,515.The optimal opening schedule is as follows:It can be seen that the buildings should be openedas soon as demand constraints allow. Further, with thegiven demand there is room for additional high rise space,and especially for garden space.It should be noted that the model is solving for aschedule of building openings.Actual construction takes6BuildingYearOpenedHigh15rise23314-252Garden63717from twelve to eighteen months and must be started accordingly.The lag time between start and opening will have no effecton the outcome of the optimal solution due to the fact thatall present