page 1page 2page 3page 4page 5page 6page 7page 8page 9page 10page 11page 12page 13page 14TIME-PHASEDREAL ESTATEDEVELOPMENT:AN INTEGER PROGRAMMING MODELbyM.CHRISTOPHER BOLENOREM 4390SENIORDESIGNINENGINEERING MANAGEMENT12 May 1981Dr. RichardS. Barr, Professor-1TABLE OF CONTENTSSUBJECTPage1.0Introduction22.0The Problem23.0Assumptions44.0Description of the Model45.0Definition of Decision Variables and Coefficients66.0The Integer Programming Model77.0The Solution Phase87.1The Relaxed Solution87.2The Integer Solution87.2.1Zone Openings87.2.2Land Sales and Development87.2.3Profit87.2.4Loans97.2.5Cash Flow98.0Conclusion9TABLES and FIGURESFigure ALand Apportionment3Table 1Cost and Demand Figures5Table 2Land Sales Schedule11Table 3Land Development Schedule121.0Introduction.This real estate development problem is an expansion and refinement of aproblem suggested by Dr. Richard B. Peiser, Assistant Professor of RealEstate and Regional Science at Southern Methodist University. The ensuinginteger programming model was originally developed but never solved byMark D. Youtsey, a senior engineering student at SMU, in 1980. The follow-ing model, however, has been further developed to reflect current costs,interest rates, inflation rates, and a clearer representation of annualcash flows.This integer programming model is a mathematical investment analysis usedto determine optimal investment decisions in the development of real estate.It utilizes a modified version of linear programming to accommodate integer(0,1) variables for an integer programming solution.2.0The Problem.The problem addressed by this model is described as follows:A real estate developer/investor owning a large tract of land wishes todetermine the optimal method of developing and selling various plots ofthis land so as to maximize profit. The development project is to extendover a 5-year period. The land is divided into four zones of differentsizes or acreages (see Figure A). Each zone is further apportioned intothree types according to use or purpose: (1) residential, (2) industrial,and (3) commercial. Thatis,each zone has a different allotment of landsuitable for each of these three purposes.2FIGURE ALAND APPORTIONMENT(in acres)1 = Residential2 = Industrial3 = Commercial3ZONE1ZONE 2TvpeAllocation TypeAllocation123Total0150150123Total3001000400300ZONE3ZONE 4TypeAllocation,TypeAllocation137514752 02100325325Total400Total600The developer is faced with fixed (startup) costs associated with opening azone for development, as well as marginal or variable costs for developingeach type of land within the zone (see Table 1). Development of the landis stimulated by annual demand and revenues from selling the developed land.In order to finance these developments the investor is given a yearly lineof credit of $5 million at 18 per cent interest per annum. A one-year loanmay be procured at the beginning of each year up to the credit limit and mustbe repaid with interest at the end of that year (no compounding). Allselling prices and costs are subject to an annual inflation rate of 10 percent.3.0Assumptions.The assumptions which have been incorporated into this model are:(1)Demand for land does not vary with development (constant demand).*(2)Price does not vary with demand (constant price with respect todemand).*(3)Once a zone is opened for development any and all parts of thatzone are eligible for development and subsequent sale.4.0Description of the Model.This model is a large integer programming model. The formulation is comprisedof a maximum profit objective function in 160 integer and continuous (linear)variables, subject to 160 constraints.The maximum profit sought in this model is the result of revenues from landsales and interest onn cash less the costs of land development and interest* These assumptions, although contradictory to economic theory, are necessaryto enable this model to take on its integer characteristics, as opposed tothe- more complex, dynamic programming model.4b1-,*An inflation rate of 8° per annum is built into each year's costs and prices.For each of the 5 operating years, this is reflected in the coefficientsof the decision variables.5* COSTS:TABLE 1I.FixedZoneStartup Costs:Cost (millions)II.Variable Costs:Land TypeCost1234* PRICES:I. Selling$1.50$1.75$2.00$2.25Prices for Land1$10,0002$ 6,0003$ 6,000TypeCost Per Acre1$30,0002$44,0003$04000DEMAND: (in acres)YearType12 3 4 51100200300300 3002 255075 75753050505050on loans. The profitability of this endeavor is constrained by:(1)land availability(2)land apportionment(3)demand for developed land(4)capital availability (due to credit limits)(5)the fact that land may not be sold until it is developed..5.0Definition of Decision Variables and Coefficients.The decision variables and coefficients for this model are as follows:LSijt=amount of land sold of development typeduring year "t" (in acres),whereLD ijtLAijDjtLtCtrtbitSitIT' in zone "i"amount of land developed of type "j" in zone "i" duringyear "t"(inacres).amount of land in zone "i" allocated for development typedemand for land of type "j" during year "t".loan procured during year "t" for investment toward thedevelopment of land.idle cash on deposit in a bank during year "t" (gaininginterest of 5.25% annually while on deposit).interest rate on any loan procured at the beginning of year18% per annum.0if zone"i"is not developed in year "t" (not opened)1if zone"i"is opened for development in year "t".fixed startup cost for developing zone "i" in year "t".6"j".nt".i = 1,2,3,4 (zone numbers)j=1land for residential development2""industrial"3""commercialitt = 1,2,3,4,5(years of operation and investment).7selling price of land type "j" in year "t".Pjtmarginal (variable) cost of developing land type "j" inMjtyear "t".(CASHIN)t=sum of all cash in-flows(CASHOUT)t=sum of all cash out-flowsduring year "t".during year "t".6.0The Integer Programing Model.This real estate development problem maybe formulated as the followinginteger programming model:Ob'ective Function:Maximize Profit = Z5_t=i=1wherei =j=t =3N(LS..j=11,2,3,41,2,31,2,3,4,5P- LD)-bsit]-ijt, jti'tMjtitt=rLt1_Isubject to:55(1)rLSij ttE=1LDijt(2)LSilt <_ Dlt(1, residential use)LSi2tZD2t(j =2, industrial use)LSi3t S D3t(,commercial use).(3)tLDijk<_IAij(,lSik)51(4)tlSit= 1wereSit= (0,1)(5)(CASHIN)t=ZLSij(t_l)Pij(t-1)+ 1.0525 Ct_l+ Lt(6)(CASHOUT)t=tLDijtMjt+rsitSit + (1 +rt-1)Lt-l(7)(CASHIN)t=