Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 151982-01 Spring 1982 SOUTHERN METHODIST UNIV The Optimal Schedule for the Opening of Buildings in an Office Complex Scot G. Andrus The Optimal Schedule for the Opening of Buildings in an Office Complex by Scot G. Andrus DEPARTMENT OF OPERATIONS RESEARCH AND ENGINEERING MANAGEMENTSCHOOL OF ENGINEERING AND APPLIED SCIENCE DALLAS, TEXAS 7275The Optimal Schedule forthe Opening of Buildings inan Office Complex by Scot G. Andrus Senior Design inEngineering Management OREM LI39O Presented 29 April, 1982Assistant Professor: Dr. Richard S. BarrCONTENTS SubjectPage The Problem2 The Model2 Definitionof the Model3 Definitionof the Varibles3 Assumptions Definitionof the Coefficients5 The FormulatedModel05 The OptimalSolution6 Conclusion7Tables I - The Optimal Rent-out Schedule9 II- Net Profit per Building .10 Ill-Demand for Office Space11 IV- Sale Value of the Buildings11 V - Net Operating Income12 VI- Construction Costs13 12 The Problem This model was formulated to analyze a problem presented by a land development firm planning to construct a group of office buildings. The developer has acquired approximately fifty two acres of land in a prime north Dallas location and is planning to construct seven office buildings over the next several years. The goal of this model is to determine the schedule of building openings which will maximize the firm's profit. The Model The model deals with two distinct types of office buildings, high rise and garden. The two types each have their own net incomes, demands for space, and construction costs. The model can, in fact, be considered as two independent models. This fact is very helpful and makes finding the optimal solution much easier, due to the size of the final formulated model. In the formulation of the model, variables are separated by building number, with buildings one, two, and three being high rise, and four, five, six, and seven Ibeing garden type. Buildings one and three are essentialy the same,each having 350k sq. ft. of available space. IBuilding two is slightly larger with 450k sq. ft. Buildings four and five are likewise essentially the same eaCh Ihaving 60k sq. ft. Buildings six and seven are also I.Definition of the Model This model as formulated is a mixed integer linear programming model with 105 variables and 85 constraints. The objective function finds the optimal solution using present value of the rent plus the present value of the sale value minus the present value of the construction cost for each of the seven buildings. The model is constrained by the demand for each type of office space, by the maximum capacity of each building, and by the obvious requirement that a building must be opened before it is rented out. Definition of Variables There are three sets of variables in this model, ZiyBiy, and SALE. Zjy - integer variable either 0 or 1. A,1 indicates that building i is to be opened in year y. A 0 indicates no change in building's status. 3 Biy - continuous variable indicating the square feet to be rented in buildi SALEI - continuous variable indicating the building i in.year 7. It is found the total number of sq. ft. rented NOli divided by the capitalization Other constraining variables:number of g j in year y sale value of by multiplying out by the rate of 10%. CCjy - construction cost for building i in year y. fly - demand for high rise space in year y D2y - demand for garden space in year . MC - Maximum capacity of building i.4 Iessentially the same with 75k sq. ft. each. The demand for each type of office space is based on projections for Dallas as a whole, and this location in particular. The actual figures were provided by the firm involved. It should be noted the firm involved is primarily involved in land development and is, therefore, not interested in owning the buildings for an extended period of time. They have specified that the buildings be sold by the model at the end of the time period, in this case seven years. Assumptions There are several basic assumptions incorporated Iintothe formulation of this model. IThefirst assumption is that once a square foot of space is rented, it will remain rented for the remainder Iofthe seven year time span.This is used so that income can be considered as a cash stream for n years, thus Ieliminatingthe need to keep track of rent for each year. IThisway it can be considered as a lump sum. The second assumption was one stipulated by the firm Iinvolved;that is, there is to be no capital restriction placed on the opening of the buildings.They have enough Icapital to cover the construction of any or all of the Ibuildingsat any time.IDefinition of Coefficients The coefficient on the Zj variable is thepreserit value of the construction cost for building i in year y' adjusted of inflation of 8% per year. The coefficient on the Bjy variable is the present value of the cash stream of rent for building i in year y' adjusted for inflation of 8% per year. The coefficient on the SALEi variable is the present value of the sale value of building i in year seven. The present value discount rate is 20% in all three cases. The Formulated Model The formulated model can be mathematically represented in the following way: Maximize Profit= y1 i1 (Pvy*Biy*oIiy+Pv7*SAIi_pvy*ziy*cciy) Subject To: 1)•iy ly BiyD2y For y from 1 to 7 2)(Zjy*Djy)Bjy or I from 1 to 7 or i from 1 to 7 3)iy- 1 11=1 'or I from 1 to 7 Zjy-1 y1 For i from 1 to 71 I I(High rise space rented less than or equal to demand) (Garden space rented less than or equal to demand) (The building must be opened before pace is rented out) (Total space leased equal to building capacity) (The building is to be opened once and only once)The Optimal Solution The formulated model was solved using the LINDO software package. LINDO was used because of its ability to optimize the model -using 0,1 variables. Due to the model's size and the increased complexity of solving an integer model, it was necessary to optimize the model in two parts. The was acomplished by setting specific Z1y values for i=1,2,3 and allowing the software to optimize for buildings four, five, six, and seven. The optimal values for Ziy i=4,5,6,7 were then set, and the software was allowed to optimize for buildings one, two, and three. The optimal value yields a present value net profit of $37,118,515. The optimal