Formulation of problem model

Figure 3.8 shows the stepwise payment schedule, which represents most cases in the practical world where payments are received at certain time periods.

However, this study treats the payment with a continuous linier function to simplify the calculation.

Figure 3.8 Contractor's Expenses and Owner's Payments (Source: PMBOOK)

In the present analysis, net present value will be used as economy measures 23

of time value of money. Future costs have to be discounted to a common point in time; say present time (or project start time). To consider the time value of money, the activity cost incurred at its scheduled finish (SF) has to be discounted by some factor. The discounted cost of an activity i can be calculated as

in which Ci = cost of activity i; SFi is its scheduled finish; and 1/(1+ r)^SFi = discount factor expressed in terms of interest rate (r). Discount factor in the exponential form, given in Eq. (1), is too complicated to be handled in a mathematical optimization model. Instead, a simplified form (but also practical) of Eq. (1) will be used. Since r is expressed as a fraction with values less than one, the discount factor (1+ r^)-SFi can be expanded in a polynomial form as

From the practical point of view, units of activities’ durations are usually days or even weeks. Also, interest rate usually has very small values compared to unity (i.e., r ≤ 1). Therefore, higher order terms of the discount factor expansion can be truncated to become

Although the discount factor given by Eq. (2) is an approximation for the exact one expressed in Eq. (1), the recorded error can be neglected. For example, if r is taken 12% annually and durations are measured in weeks, the exact value of discount factor for a time period of 50 weeks is 0.8911 {1 /[1+ (0.12/52)] X 50} while the corresponding approximate value is 0.8846 [1− (0.12 /52) X 50]

with an error value of 0.7%. For the same value of interest rate (i.e., 12%), the exact value of discount factor for a time period of 100 days is 0.967663 while the approximate value is 0.967123 (error = 0.06%). It is, therefore, very good approximation to use only the first two terms of Eq. (2) instead of using the exact formula. The compacted form of discount factor given by Eq. (2) will be adopted in the present analysis. The discounted cost of activity i given by Eq. (1) in terms of the adopted discount factor will be

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Consider a project having n activities, where utility data for project activities are represented by discrete functions. For each activity i, mi discrete points are to be specified, where mi 2 ≥ 1. Every discrete point represents a specific way of carrying out the activity. The normal (least cost) and crash (least duration) discrete points represent the two extremes of the activity time-cost function. For activities having only one discrete point, the normal and crash points coincide.

The primary information obtained from traditional scheduling are basically activities’ start and finish timings (early and late) and floats. The duration and the corresponding cost for an activity are selected optimally from their utility data to satisfy the objective function and the imposed constraints. If the start time of an activity is determined, the finish time can be specified by adding the selected activity duration, and vice versa. The SF of activities will be used in the present model formulation, as will be illustrated after.

Decision Variables

A single zero-one (binary) variable, x, is needed for each discrete point in the utility data of each activity. Let Di and Ci be the variables representing duration and cost of activity i, respectively. The duration and cost for an activity i, in terms of zero-one variables, can be expressed as follows:

in which xij = zero-one variable belongs to the discrete point number j for activity i and dij and cij = j duration and cost of activity i. Zero-one variables are introduced to ensure that only one construction method (discrete point) is selected per activity.

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Zero-One Variables Constraints

The mathematical model can be forced to select a single construction method per activity (duration and the corresponding cost) at a time for an activity i if the following condition is satisfied:

This type of constraint will be referred to as zero-one variables constraint.

Since every discrete point requires zero-one variable, the number of zero-one variables needed is the sum of discrete points for all activities, while required number of zero-one constraints equals number of project activities, n.

Network Logic Constraint

The completion time of a project could be constrained by one of two methods (Crowston 1970). The first approach is to allow for a precedence constraint for each immediate preceding relationship in the project network. This approach was used in almost all existing optimization techniques (e.g., Liu et al. 1995;

Eldosouky et al. 1991; Cusack 1985; Perera 1982; Kapur 1973; Meyer and Shaffer 1965). The second is to allow for one constraint for each path from the first activity to the last one in the project network. In the present model, the first approach will be adopted.

The logical relationship between any two consecutive activities i and its immediate predecessor, p, is expressed mathematically as

where SSi = scheduled start of activity i and SFp = scheduled finish of its predecessor p. The SF of activity i, equals its SSi plus its duration. The logical relationship constraint can, then, be expressed by Eq. (7), in which NPi is the number of preceding activities to activity i

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Project Completion Constraint (s)

Project completion is controlled by the latest finish time of ending activities. If the number of ending activities is denoted by NE, the project completion constraint(s) is given by Eq. (8), in which β is the desired project duration

SFk≤ β, k = 1,2, . . . ,NE (8)

The upper and lower bounds on A are the normal project du- ration and all-crash project duration, respectively.

Objective Function

In traditional TCTO analysis, the objective function is usually set to minimize the project cost. This objective of minimizing project cost is, also, adopted in the present model formulation. Since indirect cost increases linearly with project duration and usually represented as a single cost per time period (Ahuja 1984), project direct cost only needs to be minimized. The project direct cost is the summation of all activities’ costs expressed mathematically by

In Eq. (9), time value of activity cost incurred at different points of time (finish time of activities) is ignored. If the time value of money is to be considered, costs of activities have to be discounted to the project start time. Introducing the discounting factor expressed by Eq. (2) to the project direct cost given by Eq.

(9), it becomes

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As some terms in the objective function are the product of two decision variables, the proposed model takes the standard form of nonlinear mathematical programming. The model requires as input precedence relationship of project activities in the form of finish to start and activities discrete point utility data. Making use of these data, the project is analyzed to get both the all-normal and all-crash project durations. For each feasible project duration, the optimization model selects the optimum duration and cost for each activity. The project time-cost curve is then determined by adding the indirect cost component. Consequently, the optimum project duration can be specified, as will be illustrated by the following example project. (Ammar, 2010)

The following symbols are used in the problem formulation:

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