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Synthetic Cash Flow Model with Singularity Functions I:
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Theory for Periodic Phenomena and Time Value of Money
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By Yi Su 1, S.M.ASCE and Gunnar Lucko 2, A.M.ASCE
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America, Washington, DC 20064, email: [email protected].
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2
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Washington, DC 20064, email: [email protected].
Graduate Research Assistant, Department of Civil Engineering, The Catholic University of
Associate Professor, Department of Civil Engineering, The Catholic University of America,
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Abstract
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Precisely and efficiently calculating balances of cash flows is crucial for successful engineering
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economics analysis within construction project management. However, such balance calculation
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is challenged by mercurial conditions; e.g. individual cash flows may occur periodically, profit
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markups may be distributed evenly or unevenly, and the balance must consider the time value of
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money. These intricate phenomena can be modeled with singularity functions. Singularity
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functions perform customized case distinctions, which yields enormous modeling flexibility.
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Contributions to the body of knowledge are threefold: A new signal function is introduced to
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express the periodicity of incremental payments and compound interest in detail for both integer
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and non-integer periods. Outflows and inflows both explicitly consider the time value of money
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for accurate direct calculation of variable balances, which can identify breakeven points. All
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formulas are validated with worked examples. Future research can extend the new approach
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toward analyzing other phenomena, e.g. prompt payment discounts.
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Keywords: Cash flows; engineering economics; balance; time value of money; singularity
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functions
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1. Introduction
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Detailed cash flow management is crucial for survival in the economic marketplace and business
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success of construction participants (Lucko 2011b). Decision-makers must actively plan, monitor,
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and control cash flows. However, their diverse features and intricate interactions provide
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enormous challenges to calculating balances precisely and efficiently, which complicates timely
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and effective financial management. In it, the fundamental and foremost required parameter is
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the balance, which is defined as the difference between cash outflows and inflows (Kenley 2003).
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From the perspective of whether repetition occurs, cash flows can be categorized as periodic or
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non-periodic. For example, the periodic cash flows on construction projects include payments,
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financing fees (interest and possibly an unused credit fee), salary, tax, and retainage, whose
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amounts typically differ in each period. One-time or non-periodic cash flows include permit
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costs, mobilization, materials costs, retainage release, and liquidated damages for late completion.
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For valid comparisons, the time value of money must also be considered in the calculation,
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which threatens to make it even more complex. Ultimately, modeling, analysis, and optimization
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to support important strategic and tactic decisions, which may also need to consider scheduling
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and resource management, become arduous unless a powerful new approach is found. Therefore,
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a synthetic cash flow model is urgently need to fulfill the requirements of accuracy, efficiency,
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and flexibility in calculating balances with multiple parameters. Gaining such financial model
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could provide a solid theoretical foundation for active and beneficial cash flow management.
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2. Literature Review
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The relatively many cash outflows and few inflows of construction projects causes their profile
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to have a characteristic ‘sawtooth’ shape (Elazouni and Metwally 2005, Kenley 2003). Charging
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interest on any overdraft makes it even more difficult to model balances. Recent studies had to
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apply a decomposed chronological approach to calculate only month-end balances. However, the
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discrete, disjointed, and inflexible nature of such approaches has hampered cash flow modeling.
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Cash flows are a classic topic in construction management and have spurned numerous
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studies. Exhaustively surveying the literature is nearly impossible, but the stream of models can
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be grouped into two branches: Curve-fitting models versus cost-and-time-integration models
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(Navon 1997). Following this classification, the former often use polynomial regression to
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forecast cash flows (e.g. Jarrah et al. 2007, Khosrowshahi 2000, Kenley 2003, Khosrowshahi
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1988, Gates and Scarpa 1979, Ashley and Teicholz 1977). Among the wealth of techniques for
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cost-and-time-integration were integer programming (Elazouni 2009, Elazouni and Metwally
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2005, Elazouni and Gab-Allah 2004), linear programming (Tong and Lu 1992, Teicholz and
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Ashley 1978, Stark 1968), quadratic programming (Cattell 1987, Diekmann et al. 1982), systems
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analysis (Cui et al. 2010), tabular or spreadsheet methods (Park et al. 2005, Hegazy and Wassef
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2001, Hegazy and Ersahin 2001a, Hegazy and Ersahin 2001b, Halpin and Woodhead 1998,
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McCaffer 1979), and newly introduced singularity functions (Lucko 2013, Lucko 2011b). Such
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variety of different models resulted from their dissimilar research goals; some focus at a macro
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level, others at a micro level. Research on accurate and efficient balance calculation belongs to
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the micro level, which is where singularity functions best exhibit their capabilities. Non-periodic
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cash inflows and outflows have already been modeled using singularity functions (Lucko 2011b).
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This paper focuses on the more intricate periodic cash flows and contributes new theory
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with its balance model, which provides a foundation for solving further advanced research
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problems. It is called ‘synthetic’ to emphasize that it expresses the cash flows differently from
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traditional incremental approaches, via direct calculation, using feasible shortcuts if possible. To
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ultimately realize the goal of the synthetic cash flow model, three Research Objectives are set:
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1. Deriving a signal function to represent periodic phenomena that occur at integer periods;
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2. Creating a cash flow model to calculate its intricate balance with accuracy and efficiency;
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3. Extending the new synthetic cash flow model to correctly handle any non-integer periods.
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3. Singularity Functions Definition
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The term ‘singularity’ stems from calculus, where it means a point that is undefined or behaves
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in a discontinuous manner, e.g. a jump in value. Singularity functions are piecewise functions per
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Equation 1, wherein y and z are the independent and dependent variables. Their parameter a
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represents the cutoff value where the operator selects from two choices: Evaluate as an ordinary
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function with round brackets if y is equal to or larger than a, or set it to zero if y is smaller than a.
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Such a piecewise selection operation is analogous to making a judgment, and is symbolized by
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the pointed brackets. The parameters s and n have the meaning of scaling factor and exponent.
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Changing n will cause z(y) to exhibit the behavior of various functions, which can have linear or
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nonlinear characteristics. Note that for consistency, this paper uses a right-continuous definition,
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yet it is equally possible to use a left continuous version with appropriately modified equations.
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0

for y  a
n

z  y  s y  a  
n

 s   y  a  for y  a
(1)
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Although they are piecewise segmental, singularity functions are not limited to modeling
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only discontinuous phenomena, because their behavior at y = a depends on how the two branches
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are defined. Equation 1 is continuous if values from both braches are equal, but discontinuous if
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the branches return dissimilar values, e.g. z(y) = s · y – an is continuous at y = a if n = 1 (ramp),
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but discontinuous if n = 0 (step). Multiple terms are linked by direct addition or subtraction. A
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set of terms with different behaviors can be superpositioned to capture continuous but variable
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profiles, e.g. the well-known zigzag shape of cash flows. Furthermore, parameters s, n, or a
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could themselves be replaced by other singularity functions to generate an even more complex
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behavior, e.g. periodic sampling and compounding, which are examined in a following section.
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4. Chronological versus Synthetic Balance Calculation
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In the following sections, the values of each activity that are known are its planned start aS, finish
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aF, duration D, cost C, profit markup M, and the bill-to-payment-delay b. A shift d1 and delay d2
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change the start and finish of an activity as follows: A shift affects aS and aF, but not D. A delay
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modifies aF and D. The actual start and finish therefore are aS*  aS  d1 and aF*  aF  d1  d2 .
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This paper presents costs of activities as growing linearly over their duration, which is a
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widely used assumption in the literature (Elazouni and Metwally 2005, Halpin and Woodhead
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1997), for clarity and brevity of presenting the new theory. Note that singularity functions are
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sufficiently powerful to handle nonlinear patterns of cost growth via the exponent. However,
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such scenarios are beyond the scope of this paper and will be explored under future research.
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4.1 Chronological Balance Calculation Method Using Infinitesimal Offset ɛ
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A prior study used an infinitesimal duration ɛ to calculate month-end balances (Lucko 2011b),
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but this approach was arduous and inflexible due to having to recalculate all values if any
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parameter was changed, whereas the subsequently derived synthetic calculation method embeds
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the process steps of cost, bill, payment, interest, and balance into its singularity functions.
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Equations 2 through 5 (Lucko 2013) provide exact interest for linearly growing cost, where FVbal
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is the future value of a balance, A is the ‘principal’ that repeats at ends of integer periods, D is its
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duration, and δ = aF - aF is the partial period from aF until the next integer time. Example 1
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assumes that the activity features evenly distributed cost. For simplicity, profit markup is zero.
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The activity has a 3 mo. duration, its total cost is $300,000, bill-to-payment-delay is 1 mo., and
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the interest rate is 5%/mo. Figure 1 and all others show absolute costs and payments above the
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horizontal axis and their balance below it. For clarity, their resolution is exaggerated as only 0.1
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time units (giving steep slopes not steps), whereas actual values and ɛ are infinitesimally small.
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<Insert Figure 1 here>
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FVbal  A  1  i 

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D
FVbal  A  1  i   1 / ln 1  i 


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
FVbal  A  1  i   1  i   / ln 1  i 


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FVbal  A  i / ln 1  i 
D 

 1  i   / ln 1  i 

(2)
if balance grows in second half of one period
(3)
if balance grows in first half, then remains constant
if linearly growing balance across one period
(5)
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(4)
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Table 1 shows the three steps of the chronological method cropped to an accuracy of four
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decimals. It first determined the balance of Example 1 just before charging interest; then after an
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infinitesimal duration ɛ added said interest; and after yet another ɛ added the monthly progress
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payment per the magnification of the balance in the left dashed circle in Figure 1. This method is
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obviously correct, but awkward and tedious, which strongly restricts its modeling capabilities.
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Therefore, this method is merely applied in this paper to verify the correctness of the new model.
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<Insert Table 1 here>
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4.2 Synthetic Balance Calculation Method Using Future Values of Payment and Cost
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Ashley and Teicholz (1977) explained that the difference of cost profile minus payment profile is
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the net cash flow. One can calculate the future value of the payment minus the future value of the
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cost to yield the balance, which for negative balances will include all financing fees, i.e. interest.
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This synthetic balance calculation has only two steps per the right dashed circle in Figure 1. It
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does not require the separate interest step of the chronological method, which is more convenient
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to use. Table 2 lists balances at ends of integer periods, for which the results are equal to those in
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Table 1. Therefore, by applying the synthetic method based on modeling the future values of cost
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and payment with singularity functions (Lucko 2013), it becomes possible to directly obtain the
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balance with interest. Such approach would be particularly efficient for optimization purposes.
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<Insert Table 2 here>
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5. Cash Flow Model with Signal Functions
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Table 2 shows a conspicuous behavior of cash flows, namely periodic events. Interest is charged
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monthly, and the payee –a contractor – periodically receives progress payments. This time lag in
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cash flow models is the “difference between the time a resource is used on the site and the time it
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is paid for” (Navon 1997, p. 1056). For efficiency, such periodic events can be modeled just once
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theoretically and then multiplied with a ‘repetition operator’ to initiate each further occurrence.
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A need therefore exists to derive a ‘signal’ function for such periodic cash outflows and inflows.
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5.1 Signal Functions
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A novel signal function can model the periodic phenomena of receiving payments and charging
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interest. It is flexible in that costs can start and finish at non-integer times. Expanding the basic
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term of singularity functions per Equation 1, Equation 6 and 7 create singularity functions with
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two different rounding operators. The floor operator   gives the rounded down integer of the
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operand, while the ceiling operator   rounds up. Equation 8 is their difference. Setting s1 = s2 =
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1 and n1 = n2 = 1 and a1 + 1 = a2 gives Equation 9 that defines the desired basic signal function.
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z1  y   s1   y   a1
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z2  y   s2   y   a2
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z3  y   z1  y   z2  y   s1   y   a1
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zsignal  y    y   a    y    a  1
n1
1
0

for  y   a1


n1
for  y   a1

 s1    y   a1 
n2
(6)
0

for  y   a2


n2
for  y   a2

 s2    y   a 
n1
1
 s2   y   a2
(7)
n2
(8)
(9)
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1
Start and finish, period, and amplitude of the signal function can be modified easily via
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the respective parameters. Equation 10 expresses a unit signal starting at aS + 1 and finishing at
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aF with amplitude one. The aS and aF can be any value so that the underlying activity can start
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and finish at any time. Note that if aS or aF are non-integers, at those times the signal changes
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proportionally. This feature is valuable and can be exploited in cash flow modeling. In Example
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2 let aS* = 0.25 and aF* = 2.58. Note in Figure 2 that z1(y) = y - aS* 1 - y - aF* 1 and z2(y) =
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y - ( aS* + 1)1 - y - ( aF* + 1)1 are plotted as solid and dashed steps. The difference between
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the two step functions is the signal profile. Inserting aS* and aF* gives z1(y) = y - 0.251 - y -
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2.581 and z2(y) = y - 1.251 - y - 3.581. At time y = 1, z1(y) - z2(y) = 0.75, i.e. the signal
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strength represents the partial period from the start to the next integer. At y = 2, the difference is
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simply the full period z1(y) - z2(y) = 1. At y = 3, z1(y) - z2(y) = (2.75 - 0.42) - (1.75 - 0) = 0.58, i.e.
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the signal strength represents the remaining period from the previous signal (y = 2) to aF* = 2.58.
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zsignal  y    y   aS

1
  y   aF
1
    y    a  1
S
1
  y    aF  1
1

(10)
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<Insert Figure 2 here>
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5.2 Future Value of Payment Function
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Two tasks must be accomplished by the future value of payment function. First, calculating the
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compounded interest and second, accumulating the previous payments. The signal function can
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be multiplied with the payment function for repeated events. Equation 11 modifies Equation 10
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to the payment signal, which is delayed by b = 1 period after the bill. Continuing Example 2, aS*
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= 0.25 and aF* = 2.58. The first payment will occur at time 0.25 plus one period plus b, i.e. y =
9
1
2. The second payment adds two periods, andsoforth. The term  aF*  + b gives the time of the
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last payment, which here is 2.58 + b = 4. Equation 12 gives the current value for each payment.
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z pay _ signal  y  

*
*
 y    as  b    y    aF  b 
1
1

  y    as*  b  1   y    aF*  b  1
1
4
5
1

(11)
zeach _ pay  y  
C
 1  M   z pay _ signal  y 
D  d2
(12)
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Equation 13 applies the accounting concept of the Future Value of an annuity. However,
7
shift d1 and delay d2 may equip an activity with non-integer start or finish, so that the annuity
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assumption of equal principal in all periods is not necessarily fulfilled; the very first and last may
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differ if they cover partial start and finish periods. Equation 13 thus contains three terms: A start
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payment (potentially partial), all middle payments (full periods), and finish payment (potentially
11
partial). The control term y – (…)0 activates the partial start and finish if needed; yielding one
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only from their respective integer onward. The middle term in Equation 13 exploits that a series
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of annuity terms can be shortened by canceling each other out (Newnan et al. 2004). Collapsing
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these numerous equal payment simplifies their expression for future value. Successive activation
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of the start, middle, and finish terms yields the correct cumulative future value at any time on the
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y-axis. The principal at each payment zeach_pay is multiplied by (1+i)y – (…) to calculate its future
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value at any integer time y. This constitutes the first use of a singularity function within the
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exponent, which demonstrates the versatility of these functions in describing complex behaviors.
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zFV _ pay  y   zeach _ pay  a   b  1  1  i 
20
21

 zeach _ pay

 a   b  2  1i  1  i 
*
S
*
S

  y   a*   b  1
 S 

*
 y   aS  b 1

1
  1  i   y   a
*
 y    aS  b 1
1
*
 y    a  b
 zeach _ pay  aF*   b  1  i      F    y  aF*   b


1


*
F
0
 
 b 1

1


0
(13)
10
1
2
5.3 Future Value of Cost Function with Interest
3
Equation 14 is the signal for charging interest, altering Equation 10 by inserting the actual values
4
a* with shift d1 and delay d2. Equation 15 gives the future value of cost with compound interest.
5
Since interest is charged at the end of each month, floor and ceiling operators in Equations 13
6
and 14 ensure that the future value of payments and costs change at those ‘signal’ points in time.
7
zint _ signal  y     y   aS*

8
zcost _ int  y  
1
1
1
1
  y   aF*     y    aS*  1   y    aF*  1 

 
(14)
*
a*  1
C
1

z
 y   a  1

  1  i  int _ signal   S    1  1  i      S    y  aS*   1

D  d 2 ln 1  i   
1


0
10
1
*
*
*
0
a* 

1 z
 y    a  1
 y   a  1 
 y   a 
 1  i      S    1  i      F     1  i   1  i  int _ signal   F     1  i     F   y  aF*  



 
(15)
11
The exact interest equations for linearly growing cost are used (Equations 2 through 5) in
12
Equation 15. Same as Equation 13, it contains start, middle, and finish terms. Different from the
13
payment of Equation 12, the cost slope C / (D + d2) does not contain a profit markup, because
14
interest is charged only on balances, e.g. bank loans, not on billed cost to the payer. The common
15
factor 1 / ln(1 + i) that originated in Equations 2 through 5 can be extracted from all terms. The
16
1  i  zint _ signal  aS*  1  1 and 1  i   1  i 1 zint _ signal  a*F    in the start and finish term resemble (1 +




17
i)D - 1 in Equation 3 and [(1 + i) - (1 + i)δ] in Equation 4, respectively. For a partial start, e.g. aS*
18
= 0.25, zint_signal( aS*  + 1) returns the duration from the activity start to the next integer, matching
19
Equation 3. Yet if the activity starts at an integer, then zint_signal( aS*  + 1) gives a unit period (D =
20
1), matching Equation 5, i.e. (1 + i)D - 1 = (1 + i)1 - 1 = i. For a partial finish, 1 - zint_signal( aF* )
21
gives the duration from the activity finish to the next integer, which matches the partial period δ
1
9
1
11
1
= aF - aF in Equation 4. Yet if the activity finishes at an integer, then zint_signal( aF* ) returns a
2
unit period, again matching Equation 5, i.e. 1  i   1  i 
3
Equation 15 applies all these scenarios if start and finish time are integer or non-integer values.
4
Note that similar to Equation 13, the term with exponent zero, i.e. y - (…)0, here works as a
5
controller to prevent the double-counting of cash flows at each active signal time. If the activity
6
duration is shorter than one month, Equation 15 would need a control term to check if the partial
7
period occurs during its first half, middle, or second half. To avoid such additional complexity,
8
one can always use a finer time unit, here workdays. Developing the signal function of Equations
9
13 and 15 for various periodic phenomena in cash flows has fulfilled Research Objective 1.

1 zint _ signal a*F 
 
 = (1 + i)1 - (1 + i)1 - 1 = i.
10
11
5.4 Considering Time Value of Money
12
Ashley and Teicholz (1977) noted that contractors should apply different interest in time value of
13
money analysis; using their cost of borrowing (e.g. bank loan) during negative balances, but their
14
corporate investment rate (e.g. short-term deposits) during positive balances. The new integrated
15
model with synthetic balance calculation (rather than the decomposed chronological one) follows
16
their suggestion. Both balance calculation methods are represented in the inserts of Figure 1 for
17
better understanding. Note the characteristic zigzag shape of the cash flow profile. Said profile
18
remains parallel to the cost curve, as payments merely add upsteps. Equation 16 is a general cost
19
function without yet considering the time value of money. Equation 17 creates a stepped profile
20
(akin to billing), whose value is cost at the end of each period. Equation 18 charges interest on
21
said cost and considers time value of money. Equation 19 finally is the balance with compound
22
interest, which is a combination of Equations 13, 16, and 18 that have been derived previously.
12
C
  y  aS*
D  d 2 
1
 y  aF* 

1
zcost  y  
2
zstep_cost  y  
3
zint _ at _ signal  y   zcost_int  y   zstep_cost  y 
(18)
4
zbal  y   zFV _ pay  y    zcost  y   zint _ at _ signal  y 
(19)
1
C
   y   aS*
D  d2 
1
1
  y   aF* 

(16)
(17)
5
Equation 19 calculates the balance with interest at any time using the synthetic method.
6
Since interest is calculated only at the end of each period, the floor and ceiling operators in
7
Equations 13, 15, and 17 can only return future value of payment zFV_pay, future value of cost
8
with compound interest zcost_int, and charged interest on cost zint_at_signal at each signal time, but in
9
between remain constant, i.e. the balance profile is parallel to the cost curve, interrupted only by
10
vertical payment steps that already include the effect of interest per the second insert of Figure 1.
11
12
5.5 Considering Credit Limit
13
The credit limit is a vital factor in cash flow management. It is negotiated between contractor and
14
bank and defined as the maximum credit that that contractor can acquire for a project, including
15
“the financing interest on the most recent balance, whose sum must still be less than the credit
16
limit” (Lucko 2011b, p.523). Based on this need, the balance with interest but before adding the
17
payment can be calculated per Equation 20, which subtracts Equation 12 of each payment from
18
Equation 19 of the balance with interest. This yields the balance for checking the credit limit that
19
is shown as the solid line with interest ‘tails’ in Figure 1. Continuing Example 1, the synthetic
20
cash flow model can flexibly calculate interest in either an integrated manner per Equation 19 or
21
separately per Equation 20. In other words, the analysis can take either direction; checking if the
13
1
balance will remain within a known credit limit or determining an unknown required credit limit
2
by finding the maximum negative balance with interest. For example, a hypothetical credit limit
3
of $210,000 is insufficient, because the maximum balance is $218.0671 as listed in Table 1.
4
zbal _ check _ credit  y   zFV _ pay  y    zcost  y   zint _ at _ signal  y   zeach _ pay  y 
(20)
5
6
5.6 Calculating Breakeven Points
7
An important application of the balance model is calculating breakeven points. In Example 3, let
8
D = 2 mo., C = $200,000, aS = 0 mo., d1 = 0.25 mo., d2 = 1.3 mo. (actual start and finish are aS*
9
= aS + d1 = 0.25 mo. and aF* = aF + d1 + d2 = 3.583 mo.), b = 1 mo., interest i = 5%/mo., and M
10
= 90% of cost (purposely set high for an early breakeven). Thus its respective cost and bill slope
11
are C / (D + d2) = $60,000/mo. and C · (1 + M) / (D + d2) = $114,000/mo., respectively. Equation
12
20 inserts these values (in $1,000s) into Equation 19 to demonstrate the breakeven calculation.
13
14
15
1
1
1
1 
y 2
y 4
y 5
 1.05    1.05     66.5 1.05    y  5

0.05 
1
1
1
1
1
0
y 1
y 1
60   y  0.25  y  3.583   60 
 1.050.75  1 1.05    y  1  1.05  



ln 1.05
zbal  y   85.5 1.05 
y   2
1
 y  2  114 
0
0

1.05 
y  3
1
  1.05  1.0510.583  1.05  y  4  y  4

 
1
0
  60    y  0.25
1
1
  y   3.583  (21)

16
In Equation 21, only the term 60 · [y – 0.251-y – 3.583 1] of Equation 16 contains no
17
floor operator. All others use y, which streamlines the equation. Table 3 provides the detailed
18
calculation for the breakeven point. For example, if the y-value is within the first month {0, 1},
19
most of the terms y - (…)0 and y - (…)1 in Equation 21 are still zero, except for y – 0.251.
20
Integer y-values are substituted into zbal = -60 · y - 0.251 to check if any breakeven point exists.
21
At time y = 3 the first sign change is indentified, the balance is now positive. Then it turns
14
1
negative again during y  {3, 4}, as zbal(4) = -60 · 4 + 207.1936 < 0. Setting zbal = 0 allows
2
solving for the unknown ybreakeven = 207.1936 / (60 · y) ≈ 3.4532. Figure 3 shows the cash flow
3
profile of Example 3. Note that its balance changes from negative to positive at times 3 and 4,
4
because payments at those times are larger than the most recent balances. Balances are identical
5
to those calculated with the chronological method, e.g. at time 1 it is - 45 - 60 · [(1 + i)0.75 - 1] /
6
ln(1 + i) = -45.8335 (using Equation 3); at time 2 it is - 45.8335 · (1 + i)1 - 60 · i / ln(1 + i) + 85.5
7
= -24.1130 (previous balance compound for one period plus growth over one period per
8
Equation 5 plus payment at time 2), andsoforth. Identifying breakeven points supports cash flow
9
management in two ways: First, decision-makers could explore the impact of individual budget
10
parameters, e.g. unit cost, markup, bill period, and bill-to-payment-delay, and schedule
11
parameters, e.g. start, finish, duration shift or delay, and productivity, which facilitates actively
12
controlling of the balance. Second, the breakeven point is a watershed in creating strategies: For
13
significantly negative balances, cash flow management would seek to strengthen positive factors,
14
e.g. front-loading or prompt payment discounts, and closely monitor financing terms and the
15
credit limit, while positive periods afford more flexibility in applying strategies such as lowering
16
prompt payment discounts, and – as part of portfolio management – investing such earnings.
17
The synthetic cash flow model can efficiently calculate the balance at any time and also
18
facilitates the credit limit and breakeven analysis. Results from both balance calculation methods
19
are identical, which verifies the accuracy of the new model and fulfills Research Objective 2.
20
21
<Insert Table 3 here>
22
23
<Insert Figure 3 here>
15
1
2
6. Extended Cash Flow Model for Non-Integer Periods
3
In practice, the period of cash flows may occur in non-integer multiples of months, e.g. payments
4
from the owner may only be received after 45 days or longer. Therefore the signal function must
5
support integer and non-integer period durations. Equation 11 can only handle integers within the
6
synthetic cash flow model; it is expanded to a general form as described in the following section.
7
8
6.1 Signal Functions with Non-Integer Bill and Payment Periods
9
Payments are initiated by bills. The contractor sends a bill to the owner with a periodicity of p.
10
After yet another bill-to-payment-delay period b the owner issues the payment. This differs from
11
Equation 11, where p and b both are simply equal to one month. Thus it is necessary to explore
12
how they impact the periodicity of receiving payments. While b may fluctuate in practice when
13
transactions are processed, for planning purposes it is assumed to have a known regular period.
14
In Example 4, let D = 3 mo., C = $300,000, aS = 0 mo., d1 = 0 mo., d2 = 0 mo., and M = 0%.
15
Figure 4 shows three scenarios with different relations of p and b, with vertical solid black bills
16
and dashed gray payments. Figure 4a shows the simple case if both values are equal (p = 1 mo., b
17
= 1 mo.), neither dominates. In Figure 4b, the bill period exceeds the bill-to-payment-delay (p =
18
1.5 mo., b = 1 mo.). Here the payment period is the same as the bill period; the first payment is at
19
time 2.5 mo. This relation is switched in Figure 4c (p = 1 mo., b = 1.5 mo.) and the payment
20
period is again the same as the bill period; the first payment is at time 2.5 mo. The payment
21
period equals the bill period and is unrelated with the bill-to-payment-delay. The value b only
22
affects the first payment. The rhythm of the payment signal follows the rhythm of the bill.
23
16
1
<Insert Figure 4 here>
2
3
Equation 22 is the general signal function for any integer or non-integer bill period. The
4
bill period p in the denominator controls the cycle time of the bill signal. Equation 23 is the
5
general payment signal for any period. Its only difference to Equation 22 is subtracting b from
6
every y, which has the effect of shifting the signal profile by b time units to the right. Table 4
7
lists results from evaluating both signal functions for different inputs to verify their correctness.
8
9
10
1
  y   a*  1  y   a*  1    y   a*  1  y   a*
 
s
s
F
F
zbill _ signal  y               1     1 
  p  p 
 p   p     p   p
 p  p
 


(22)
1
  y  b   a*  1  y  b   a*  1    y  b   a*  1  y  b   a*
 
s
s
F
F
z pay _ signal  y    
    p  p    p  p  1   p  p  1 
  p   p

     
 

 
 


(23)
11
12
<Insert Table 4 here>
13
14
6.2 Cash Flow Model with Non-Integer Bill and Payment Periods
15
Expanding Equation 23 to Equation 24 gives the present value of each payment. The factor p
16
scales the cost slope with profit markup C · (1 + M) / (D + d2) for any partial periods. Equation
17
25 is the future value of payment, whose first and second payment terms act at times  aS*  + b +
18
p, and  aS*  + b + 2 · p, andsoforth. The last payment is given by Equation 26, whose term  aF* /
19
p –  aF* / p0 checks if aF* is divisible by p. If so, it returns one, the finish is the bill time itself,
20
and the last payment occurs at aF* + b. Else it return zero, the finish is not the bill time, and the
21
last payment occurs at aF* + b + ( aF* / p - aF* / p) · p. The bracketed part of that term is the
17
1
duration from aF* to the next bill time, plus aF* and b, to give the last payment time. Note that
2
Equation 25 lists middle payments between the first and last one separately. The collapsing of
3
Equation 13 cannot be applied to non-integer periods, because partial periods must be
4
compounded to the next integer time, instead of treating merely a series of identical annuities.
5
While Equation 25 appears rather lengthy, it can directly calculate the future value of payment.
6
7
8
zeach _ pay  y  
C  (1  M )
 p  z pay _ signal
D  d2
(24)
*
 y   a  b  p
zFV _ pay  y   zeach _ pay  aS*   b  1 p  1  i      S    y  aS*   b  1 p



1

*
 y   a  b  2 p 
 zeach _ pay  aS*   b  2  p  1  i      S 
 y   aS*   b  2  p

9


1
*
 y   a  b 3 p 
 zeach _ pay  aS*   b  3  p  1  i      S 
 y   aS*   b  3  p

10


1
0


0
0
11
 zeach _ pay  ylast _ pay   1  i  
 y   ylast _ pay
12
13
ylast _ pay
a*  a* 
 F  F 
p  p
0
1
 y  ylast _ pay
a*  a* 
  aF*  b   1  F   F 
p  p
0
1
0

  a*  a*
  aF*  b    F   F

 p  p
(25)
 
  p
 
(26)
14
To verify Equation 26, in Example 5, let D = 2 mo., C = $200,000, aS = 0 mo., d1 = 0.25
15
mo., d2 = 1.3 mo. (actual start and finish are aS* = aS + d1 = 0.25 mo. and aF* = aF + d1 + d2 =
16
3.583 mo.), i = 5%/mo., and M = 0%. Using p = 1 mo., b = 1.5 mo. in Equation 27 gives the last
17
payment time as 5.5 mo., which is correct, as the last bill for the 3.3 mo. activity is sent at y = 4
18
mo. and paid after the bill-to-payment-delay of 1.5 mo. For the future value of cost, since the
19
charging interest never changes, Equation 15 that tracks cost with compound interest remains the
18
1
same. Together these equations generate Figure 5, whose three scenarios of bill period and bill-
2
to-payment-delay relations are applied to Example 5, respectively, mirroring the theoretical ones
3
as depicted in Figure 5. The ability to handle non-integer periods fulfills Research Objective 3.
4

  3.583  3.583  
ylast _ pay  0   3.583  1.5  1 3.583  1.5   
  1  1  5.5
 1 
 

(27)
5
6
<Insert Figure 5 here>
7
8
7. Validation Example with Variable Cash Flows
9
The cash flow example introduced by Elazouni and Metwally (2005) is reanalyzed to validate the
10
synthetic cash flow model. Lucko (2011b) modeled this example with singularity functions after
11
applying them to linear scheduling, yet did not focus on periodic phenomena and their challenges,
12
nor examined issues such as the credit limit or breakeven points. Table 5 lists the example inputs.
13
14
<Insert Table 5 here>
15
16
Direct cost in this example is linearly distributed over each activity and is billed and paid
17
periodically. Indirect costs may occur periodically or not per Table 6 and could be proportionally
18
allocated to direct cost by “[a]n escalation factor E was calculated as direct cost · 1.15OH · 1.05mob
19
· 1.02tax · 1.2PR · 1.01bond = direct · 1.4927598” (Lucko 2011b, p. 527). Such simplified method
20
would be equivalent to changing the profit markup from 20% to 49.27598% in the synthetic cash
21
flow model. However, because some indirect cost only occurs once at the project start, the
22
accuracy of such approximation does not meet the objectives of this paper. Singularity functions
23
enable precise modeling and integrating cash flows with schedules, as Lucko (2011b)
19
1
demonstrated. Cash flows that occur only once or non-periodically may be modeled as an infinite
2
period in their signal function. Any cash flow can thus theoretically be treated as periodic for the
3
purposes of the synthetic cash flow model. Details of converting such non-periodic into periodic
4
phenomena are beyond the scope of this paper and are planned to be explored in future research.
5
6
<Insert Table 6 here>
7
8
Retainage is withheld at 5% off each payment and released with the last one per the
9
signal function of Equation 28. Note that this equation can be used for two purposes, as a factor
10
to multiply with the future payment of Equation 13 to subtract the retainage from each payment
11
or for its opposite, to record the total amount of retained funds that will be released eventually.
12
13
1
  y  b   a*  1  y  b   a*  1    y  b   a*  1  y  b   a*
 
s
s
F
F
zret _ signal  y   r   
    p  p    p  p  1   p  p  1 
  p   p

     
 

 
 


(28)
14
The synthetic model can express the cash flow of each activity. Its variables capture how
15
each active behaves, which allows flexible ‘what if’ analysis in the style of Figure 6, which adds
16
their individual contributions to the total profile and integrates it directly with the linear schedule
17
that is underlying this example. Such additive characteristic is provided by the Distributive Law,
18
per which one can either first add values and then multiply them with a factor or equivalently
19
one can add individual products. A ‘what if’ analysis can systematically or randomly vary these
20
parameters, e.g. the shift d1 and delay d2 of any activity and determine their impact on cash flows
21
and maximize the profit. In the synthetic model, the nature of such factor is very customizable, it
22
can be a signal function for payment, charging interest, and retainage, while the values include
20
1
the cost slope, duration, start and finish, and profit markup. This unique flexibility will facilitate
2
extending the synthetic cash flow model to new applications beyond those described in this paper.
3
For example, modeling the initial scenario of activity A per Table 5 with Equations 11
4
through 19, 21, and 28 yields Equations 29 to 38. Summing them and those of the other activities
5
generates the total cash flow profile per Figure 6d, which shows the curves cost, cumulative
6
payment, non-cumulative payment (i.e. the payment signal), balance, and the breakeven point.
7
Changing the shift d1 and delay d2 in any activity will both alter the linear schedule and its cash
8
flow profile per Figure 6, shown here for d1A = 1 mo. and for d1B = 0.25 mo. and d2B = 1.3 mo.
9
Using this synthetic cash flow model, decision-maker can efficiently and effectively explore the
10
interactions of cash flows and the underlying schedule, which enhances cash flow management.
11
z pay _ signal  y  A   y   1   y   2
12
zeach _ pay  y  A  100  1  20%     y   1   y   2

13
zFV _ pay  y  A  100  120%     y   1   y   2


1
1
    y  2


1
1
1
1
  y   3
1
1

    y   2
(29)
1
  y   3
1

  y   2   y   3   10.8%
1
14
1
(30)
 y   2
1
 y2
0
(31)
15
1
1
1
1
zint _ signal  y  A    y   0   y   1     y   1   y   2 

 

16
1
  y   0 1   y  1 1     y  1 1   y   2 1 
1
0

 y  1
 
 
   

zcost _ int  y  A  100
  10.8%   
 1  10.8%     y1 (33)
ln 1  0.8%   

17
1
1
zcost  y  A  100   y  0  y  1 


(34)
18
1
1
zstep _ cost  y  A  100    y   0   y   1 


(35)
(32)
21
1
  y   0 1   y  1 1     y  1 1   y   2 1 
1

 y  1
 
 
   
 1  1  0,8%   
  1  0.8%   
 y1

ln 1  0.8%   
1
zint _ at _ signal  y  A  100 
2
1
1
100    y   0   y   1 


3
zbal  y  A  100  1  20%     y   1   y   2

4
1
  y   0 1   y  1 1     y  1 1   y   2 1 

1
1
1


 y  1

 
 
   
 1  10.8%   
 100 y0  y1   100
  1  0.8%   
 y1



ln 1  0.8%   


5
1
1
100    y   0   y   1 


(37)
6
1
1
1
1
zret _ signal  y  A  5%    y  1 0   y  11     y  1 1   y  12 

 

(38)

(36)
1
1

1
  y   2   y   3
1


  1  0.8%   y  2  y  2

1
7
8
9
0
The new model based on singularity functions offer several advantageous features, which
are expected to support its future use in professional practice by construction project managers:
10
 Its continuous mathematical expressions can accurately calculate balances and interest at
11
any point in time in a more efficient manner than previous studies, whose models were
12
disjointed, because they were not connected to an underlying scheduling model; discrete,
13
because they could only calculate cash flows at integer points in time, and even then had
14
to do so via an extra infinitesimal time step; and inflexible, because their model mimics a
15
manual calculation that proceeds iteratively, instead of having been fully parameterized;
16
 It has introduced the signal function that expresses periodicity, which commonly occurs
17
in financial phenomena, such as cash flows of construction projects with their monthly
18
bills, progress payments, and interest charges. A brief term within this signal function
19
efficiently parameterizes the range, amplitude, and period of such repeated transactions;
20
 It is aligned with prior research on linear scheduling with singularity functions (Lucko
21
2011b). A schedule provides the fundamental time structure of construction projects. The
22
0
0
1
model thus allows an integrated analysis of time-cost behaviors, including the financial
2
impact of shifts and delays in activity starts or finishes, comparing ‘what-if’ scenarios,
3
considering the time value of money, determining the required credit limit, identifying
4
the breakeven point, and optimizing prompt payment discounts (Su and Lucko 2014);
5
 Its mathematical formulation of the new fundamental theory of cash flows is provided as
6
a ‘white box’ to lend itself to being implemented in any form of computer application;
7
 It is expandable if more constraints or factors are desired to be added, e.g. withholding a
8
retainage from payments or linking the model to resource use (Lucko 2011a) to achieve
9
a truly multi-objective optimization. In fact, any parameters in the model could be
10
expanded as functions of time, e.g. having the markup for profit time-dependent to
11
replicate unbalanced bidding or letting interest change over time for additional realism;
12
 In essence, it is a mathematical way to define the rules for efficient and effective
13
calculation. It can be used for both planning and managing, as it is indifferent to the
14
desired purpose of its input and output data. Thus a decision-maker could use it for a
15
what-if analysis while modifying a schedule to assess how cash flow may be impacted.
16
Or the model could be applied to data from project records for either retroactive analysis
17
of what happened or to forecast the anticipated cash flows for the remainder of the
18
project. It can also calculate the credit limit or find if and when it would be exceeded,
19
and how much interest is paid in a specific period or total. Comparing such scenarios
20
allows assessing if the project is currently over budget or will exceed it in the future.
21
22
<Insert Figure 6 here>
23
23
1
8. Conclusions and Contributions to the Body of Knowledge
2
This paper has commenced by reviewing previous studies on cash flow modeling and has found
3
that a new cash flow model is needed to adequately express the intricate cash flows and calculate
4
their balance with both accuracy and efficiency. Accuracy is needed because decision-making
5
for cash flow management relies upon a model that can correctly generate the complete balance
6
of different scenarios for comparisons. Efficiency is needed because cash flows form a system of
7
considerable complexity, whose elements interaction in different ways to generate the overall
8
characteristic profile. The new synthetic cash flow model has proven its modeling and analytic
9
capabilities, which are afforded by singularity functions, to accomplish these overarching goals.
10
Three Research Objectives have been fulfilled. First, special and general forms of signal
11
functions have been developed to model periodic phenomena in cash flows such as e.g. progress
12
payment and charging interesting. Second, these signal functions have been employed in a new
13
synthetic cash flow model that is based on singularity functions, automatically considers the time
14
value of money, and can calculate breakeven points for the profile. Comparing it with previous
15
approaches and the chronological method of calculating balances at specific times only that had
16
to be separated by an infinitesimal duration ɛ , the new model has the virtue of fulfilling both the
17
goals of being accurate and effective. It has been verified with several examples, has been made
18
transparent by providing its entire formulation in mathematical equations, and has been validated
19
by re-analyzing a previously published exemplar for cash flows from the literature, which it has
20
been able to replicate in detail, yet due to its integrated formulations in a more flexible manner.
21
Contributions to the Body of Knowledge of engineering economics within construction
22
project management include that it is now possible to not only accurately calculate the balance of
23
complex cash flows scenarios, but also do so comprehensively and continuously – in other words,
24
1
with a single expression for each step of the cost, bill, payment, and financing process. The new
2
synthetic cash flow model is superior to the old chronological approach of calculating a balance,
3
because it directly embeds bill period, bill-to-payment delay, retainage, interest, credit limit, and
4
TVM into the singularity functions. It is also seamlessly connected with the underlying linear
5
scheduling method, thus gaining a fully integrated treatment of time and cost. Another virtue of
6
the new model is that singularity functions terms activated one by one only with the passage of
7
time, i.e. an increasing y-value. All terms beyond the current cutoff still remain at zero and do
8
not even need to be evaluated. In other words, the actual calculation is in most cases shorter than
9
the full expression. Yet another virtue is that it mathematically captures any user-defined rules.
10
Substituting any time value or other parameter input into its equations, the model will return the
11
cash flow results for precisely that condition. Following Ockham’s razor of being ‘as complex as
12
necessary, but as simple as possible’, the model possesses a mathematical elegance by keeping
13
each equation compact, which enhances its clarity, verifiability, and supports a future computer
14
implementation. Its equations can be easily implemented in any computer code or evaluated with
15
spreadsheets. While it is anticipated that future real-world use will rely on such computerization,
16
which will greatly improve the speed of performing calculations, the model will always retain its
17
inherent transparency and the feature that it could be completely evaluated by hand, if needed. Its
18
extensibility equips practitioners and researchers in construction management with a powerful
19
tool to explore and apply even further decision-making concepts as described in the final section.
20
21
9. Recommendations for Future Research
22
The accuracy and efficiency of the synthetic cash flow model enables several avenues of further
23
investigation in financial management, including, but not limited to, prompt payment discounts
25
1
and unbalanced bidding scenarios or unevenly distributed markup, whose goals are to achieve an
2
earlier breakeven point or maximize profit. For such questions of timing of activities and their
3
related financial events the explicit inclusion of shift d1 and delay d2 is particularly advantageous,
4
because it enables ‘what if’ analysis and implementing optimization techniques. Moreover, the
5
various performance measures that are used in the earned value method could be converted into
6
singularity functions and could consider the time value of money. Furthermore, future research
7
could explore the interaction of cash flows with resource leveling or allocation and with time-
8
cost-tradeoff for multiobjective optimization, which would integrate the new synthetic cash flow
9
model even further with scheduling techniques and resource utilization. All of these phenomena
10
are subject to the already described conditions and constraints of cash flows, e.g. the credit limit.
11
12
10. Acknowledgement
13
The support of the National Science Foundation (Grant CMMI-0927455) for portions of the
14
work presented here is gratefully acknowledged. Any opinions, findings, and conclusions or
15
recommendations expressed in this material are those of the authors and do not necessarily
16
represent the views of the National Science Foundation.
17
18
11. References
19
Ashley, D. B., Teicholz, P. M. (1977). “Pre-estimate cash flow analysis.” Journal of the
20
Construction Division 103(CO3): 369-379.
21
Cattell, D. W. (1987). “Item price loading.” Proceedings of the International Congress on
22
Progress in Architecture, Construction and Engineering, Johannesburg, South Africa,
23
July 21-23, 1987, Institute of South African Architects, Randburg, South Africa, 2: 1-20.
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1
2
3
4
Cui, Q., Hastak, M., Halpin, D. W. (2010). “Systems analysis of project cash flow management
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Diekmann, J. E., Mayer, R. H., Stark, R. M. (1982). “Coping with uncertainty in unit price
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5
Elazouni, A. M., Gab-Allah, A. A. (2004). “Finance-based scheduling of construction projects
6
using integer programming.” Journal of Construction Engineering and Management
7
130(1): 15-24.
8
Elazouni, A. M., Metwally, F. G. (2005). “Finance-based scheduling: Tool to maximize project
9
profit using improved genetic algorithms.” Journal of Construction Engineering and
10
11
12
13
14
15
16
Management 131(4): 400-412.
Elazouni, A. M. (2009). “Heuristic method for multi-project finance-based scheduling.”
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Gates, M., Scarpa, A. (1979). “Preliminary cumulative cash flow analysis.” Cost Engineering
21(6): 243-249.
Halpin, D. W., Woodhead, R. W. (1998). Construction management. Taylor and Francis, New
York, NY.
17
Hegazy, T., Ersahin, T. (2001a). “Simplified spreadsheet solutions. I: Subcontractor information
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system.” Journal of Construction Engineering and Management 127(6): 461-468.
19
Hegazy, T., Ersahin, T. (2001b). “Simplified spreadsheet solutions. II: Overall schedule
20
21
22
optimization.” Journal of Construction Engineering and Management 127(6): 469-475.
Hegazy, T., Wassef, N. (2001). “Cost optimization in projects with repetitive nonserial
activities.” Journal of Construction Engineering and Management 127(3): 183-191.
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1
Jarrah, R., Kulkarni, D., O’Connor, J. T. (2007). “Cash flow projections for selected TxDoT
2
highway projects.” Journal of Construction Engineering and Management 133(3): 235-
3
241.
4
5
Kenley, R. (2003). Financing construction: Cash flows and cash farming. Spon Press, Taylor
and Francis, New York, NY.
6
Khosrowshahi, F. (1988). “Construction project budgeting and forecasting.” Transactions of the
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Annual Meeting, New York City, NY, July 10-13, 1988, American Association of Cost
8
Engineers International, Morgantown, WV: C.3.1-C.3.5.
9
Khosrowshahi, F. (2000). “Information visualization in aid of construction project cash flow
10
management.” Proceedings of the 4th International Conference on Information
11
Visualization, eds. Banissi, E., Bannatyne, M., Chen, C., Khosrowshahi, F., Sarfraz, M.,
12
Ursyn, A., London, Great Britain, July 19-21, 2000, IEEE Computer Society,
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Washington, DC: 583-588.
14
Lucko, G. (2011a). “Integrating efficient resource optimization and linear schedule analysis with
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singularity functions.” Journal of Construction Engineering and Management 137(1): 45-
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17
Lucko, G. (2011b). “Optimizing cash flows for linear schedules modeled with singularity
18
functions by simulated annealing.” Journal of Construction Engineering and
19
Management 137(7): 523-535.
20
Lucko, G. (2013). “Supporting financial decision-making based on time value of money with
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singularity functions in cash flow models.” Construction Management and Economics
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31(3): 238-253.
23
McCaffer, R. (1979). “Cash flow forecasting.” Quantity Surveying August: 22-26.
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1
Navon, R. (1997). “Cash-flow forecasting and management.” Proceedings of the 5th
2
Construction Congress, ed. Anderson, S. D., Minneapolis, MN, October 5-7, 1997,
3
American Society of Civil Engineers, Reston, VA: 1056-1063.
4
Newnan, D. G., Eschenbach, T. G., Lavelle, J. P. (2004). Engineering economic analysis. 9th ed.,
5
Oxford University Press, New York, NY.
6
Park, H. K., Han, S. H., Russell, J. S. (2005). “Cash flow forecasting model for general
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contractors using moving weights of cost categories.” Journal of Management in
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Engineering 21(4): 164-172.
9
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11
Su, Y., Lucko, G. (2014). “Synthetic cash flow model with singularity functions II: Analysis of
12
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13
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14
Teicholz, P., Ashley, D. (1978). “Optimal bid prices for unit price contract.” Journal of the
15
16
Construction Division 104(CO1): 57-67.
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17
provided quantities.” Construction Management and Economics 10(1): 69-80.
18
19
Notation
20
Symbols
21
a
=
cutoff in singularity function;
22
A
=
principal paid periodically;
23
b
=
bill-to-payment delay;
29
1
C
=
total cost of an activity;
2
d
=
variable affects start or finish of an activity (shift or delay);
3
D
=
scheduled duration of activity;
4
FV
=
future value;
5
i
=
interest in percent of balance per period;
6
k
=
thousands of dollars;
7
M
=
planned profit markup in percent of cost;
8
n
=
exponent in singularity function;
9
p
=
bill period;
10
s
=
scaling factor in singularity function;
11
y
=
independent variable along horizontal axis, here time;
12
z
=
dependent variable along vertical axis, here money;
13
ɛ
=
infinitesimal duration;
14
δ
=
partial period from activity finish to next integer time;
15

=
belong to set;
16

=
empty set;
17
 
=
brackets of singularity functions;
18
 
=
floor operator rounding downward to integer;
19
 
=
ceiling operator rounding upward to integer.
=
balance function with considering time value of money;
20
21
Subscripts
22
bal
23
bal_check_credit =
balance function for checking credit limit;
30
1
bal_disc
=
2
balance function with considering time value of money on prompt
payment scenario;
3
bill_signal
=
signal at each bill time;
4
breakeven
=
breakeven point;
5
ceiling
=
upper limit of the discount;
6
cost
=
cost function without considering time value of money;
7
cost_int
=
future value of cost with compound interest
8
each_pay
=
payment amount at each payment time;
9
floor
=
lower limit of the discount;
10
F
=
finish of activity;
11
FV_pay
=
future value of accumulated payment;
12
int
=
interest;
13
int_at_signal =
charged interest on cost at each signal time;
14
int_signal
=
signal at each charging interest time;
15
last_pay
=
last payment;
16
payee
=
participant who sends bill and may offer discount;
17
payer
=
participant who pays bill and may take discount;
18
pay_signal
=
signal at each payment time;
19
ret_signal
=
signal for retainage;
20
step_cost
=
step function of cost which return the cumulated cost only at the end of
21
each period;
22
signal
=
signal term;
23
S
=
start of an activity;
31
1
1
=
index for time shift or counting index;
2
2
=
index for time delay or counting index.
=
variable that includes shifts and/or delays.
3
4
Superscripts
5
*
32
Table 1
Click here to download Table: Paper 1 table 1.doc
Table 1: Balance with Interest Using Infinitesimal Offset ɛ
y
Cost
[months] [$1,000]
1
-100
Interest
Payment
Balance
[$1,000]
[$1,000]
[$1,000]
0
0
-100.0000
1+ɛ
0
100 · i / ln(1 + i) – 100 = 2.4797
0
-102.4797
1 + 2ɛ
0
0
0
-102.4797
-100
0
0
-202.4797
0
-210.0833
2
2+ɛ
0
102.4797 · i + 100 · i / ln(1 + i) - 100 = 7.6037
2 + 2ɛ
0
0
100
-110.0833
-100
0
0
-210.0833
0
-218.0671
3
3+ɛ
0
110.0833 · i + 100 · i / ln(1 + i) - 100 = 7.9838
3 + 2ɛ
0
0
100
-118.0671
4
0
0
0
-118.0671
4+ɛ
0
118.0671 · i = 5.9034
0
-123.9705
4 + 2ɛ
0
0
100
-23.9705
5
0
0
0
-23.9705
5+ɛ
0
23.9705 · i = 1.1985
0
-25.1690
5 + 2ɛ
0
0
0
-25.1690
Table 2
Click here to download Table: Paper 1 table 2.doc
Table 2: Balance with Interest Using Future Values of Payment and Cost
y
Future Value of Payment (Step 1) Future Value of Cost (Step 2)
[months]
1
[$1,000]
0.00
[$1,000]
100.00
4
100 · (1 + i)1 + 100
100 · [(1 + i)3 – 1] / ln(1 + i)
= 205.00
= 323.0671
-110.0833
-118.0671
100 · (1 + i)2 + 100 · (1 + i)1 + 100 323.0671 · (1 + i)1 = 339.2205
= 315.25
5
-102.4797
100 · [(1 + i)2 – 1] / ln(1 + i)
= 210.0833
3
[$1,000]
100 · [(1 + i)1 – 1] / ln(1 + i)
= 102.4797
2
Balance
315.25 · (1 + i)1 = 331.0125
-23.9705
339.2205 · (1 + i)1 =356.1815
-25.1690
Table 3
Click here to download Table: Paper 1 table 3.doc
Table 3: Breakeven Point Calculation
y
Events
Equation 21
Balance Analysis
Breakeven
[months]
[-]
[$1,000]
[$1,000]
[-]
{0, 1}
Start at y = zbal = -60 · y – 0.251
zbal = 0, (y ≤ 0.25)
0.25
-45 < zbal < 0, (0.25 < y < 1)
No
1
-
zbal = -45.8335
-45.8335
No
{1, 2}
-
zbal = -60 · y + 14.1665
-105.8335 < zbal < -45.8335
No
-24.1130
No
-84.1130 < zbal < -24.1130
No
2
{2, 3}
1st Payment zbal = -24.1130
-
zbal = -60 · y + 95.8871
3
2nd Payment zbal = 27.1936
27.1936
Yes
{3, 4}
Finish at y = zbal = -60 · y + 207.1936
-7.8064 < zbal < 27.1936
Yes
3.583
(let zbal = 0, solve for y)
y≈
3.4532
4
{4, 5}
5
3rd Payment zbal = 106.3213
106.3213
Yes
zbal = 106.3213
106.3213
No
4th Payment zbal = 178.1374
178.1374
No
-
{5, 6}
-
zbal = 178.1374
178.1374
No
6
-
zbal = 187.0443
187.0443
No
Table 4
Click here to download Table: Paper 1 table 4.doc
Table 4: Signal Function Verification
(a) bbill = b = 1
(b) bbill = 1.5, b = 1
(c) bbill = 1, b = 1.5
y zbill_signal(y) zpay_signal(y) zbill_signal(y) zpay_signal(y) zbill_signal(y) zpay_signal(y)
0.0
0
0
0
0
0
0
0.5
0
0
0
0
0
0
1.0
1
0
0
0
1
0
1.5
0
0
1
0
0
0
2.0
1
1
0
0
1
0
2.5
0
0
0
1
0
1
3.0
1
1
1
0
1
0
3.5
0
0
0
0
0
1
4.0
0
1
0
1
0
0
4.5
0
0
0
0
0
1
5.0
0
0
0
0
0
0
Note: aS* = 0 mo, D = 3 mo.
Table 5
Click here to download Table: Paper 1 table 5.doc
Table 5: Activity List and Direct Cost for Initial Configuration
(adapted from Elazouni and Metwally 2005, Lucko 2011)
Name
[-]
D
aS
aF
Predecessor Successor Unit Cost
[months] [month] [month]
[-]
[-]
C
[$/month]
[$]
A
1
0
1
Start
C, D, F
100,000
100,000
B
2
0
2
Start
C, E
105,000
210,000
C
1
2
3
A, B
Finish
110,000
110,000
D
1
1
2
A
Finish
105,000
105,000
E
3
2
5
B
Finish
115,000
345,000
F
2
1
3
A
Finish
105,000
210,000
Note: d1 = d2 = 0; bbill = 1 mo.; b = 1 mo.; i = 0.8%/mo.; and M = 20%.
Table 6
Click here to download Table: Paper 1 table 6.doc
Table 6: Indirect Cost Items
(adapted from Elazouni and Metwally 2005, Lucko 2011)
Item
Basis
Percent
Frequency
[-]
[-]
[% of basis]
[-]
Overhead
Direct
15
Periodic
Mobilization Direct + OH
05
Non-periodic
Tax
Direct + OH + mob
02
Periodic
Profit
Direct + OH + mob + tax
20
Non-periodic
Bond
Direct + OH + mob + tax + Profit
01
Non-periodic
Figure 1
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z(y) [$1,000]
Cost
Payment
y [months]
Balance
ε
ε
Chronological
Synthetic
Figure 1: Cash Flow Profile for Example 1
Figure 2
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z(y) [unit]
z1(y)
z2(y)
zsignal(y)
y [time]
Figure 2: Example of Signal Function
Figure 3
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z(y) [$1,000]
Payment
Cost
y [months]
Balance
Breakeven
Figure 3: Cash Flow Profile for Example 3
Figure 4
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(a): bbill = b
(b): bbill > b
(c): bbill < b
Figure 4: Three Scenarios for Bill and Payment Periods
Figure 5
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(a): bbill = b = 1
(b): bbill (1.5) > b (1)
(c): bbill (1) < b (1.5)
Figure 5: Three Scenarios for Bill and Payment Periods for Example 5
Figure 6
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(a): Initial Schedule
(b): d1A = 1 mo.
(c): d1B = 0.25mo.,
d2B = 1. 3 mo.
(d): Cash Flow of (a)
(e): Cash Flow of (b)
(f): Cash Flow of (c)
Figure 6: Cash Flow Profiles for ‘What-If’ Analysis
Figure Caption List
1
List of Tables
2
Table 1: Balance with Interest Using Infinitesimal Offset ɛ
3
Table 2: Balance with Interest Using Future Values of Payment and Cost
4
Table 3: Breakeven Points Calculation
5
Table 4: Signal Function Verification
6
Table 5: Activity List and Direct Cost for Initial Configuration
7
(adapted from Elazouni and Metwally 2005, Lucko 2011)
8
Table 6: Indirect Cost Items
9
(adapted from Elazouni and Metwally 2005, Lucko 2011)
10
11
List of Figures
12
Figure 1:Cash Flow Profile for Example 1
13
Figure 2: Example of Signal Function
14
Figure 3: Cash Flow Profile for Example 3
15
Figure 4: Three Scenarios for Bill and Payment Periods
16
Figure 5: Three Scenarios for Bill and Payment Periods for Example 5
17
Figure 6: Cash Flow Profiles for ‘What-If’ Analysis
1