Homogeneous Works in WBS for PDCA/EVM Analysis

Proper definition of work packages in WBS with appropriate analytic cost functions is itself the most formidable assignment in PDCA management of construction projects. The simplest form of WBS definition is characteristic to (linear and) homogeneous works. Homogeneous works are such that resources are scalable during the control period to achieve a proportionate scaling of work value in that period. That is kDV = f (kc,r,t).Dt.

Homogeneous works are linear without an intercept. The cost function can be expressed as a linear function DV = c.rDt + m; but m=0. The intercept m is a fixed value of work associated with start of work. Homogeneous cost functions are integrable, scalable, invertible and monotonously increasing. At the end of work, measurement function Vb = Sum of c.r [Tfj - Tsj] {for j = 1 to p} if there are (p-1) breaks in execution of work. [Note that p=1 for fixed duration works.] Work executed in ith segment of time can be expressed as a fraction of total measurement function as follows:
Vi = c.r [Tfi - Tsi] = [Tfi - Tsi]/ Sum of {[Tfj - Tsj] for j = 1 to p} Vb = ti Vb {for ti less than 1}

Noting that if Vb is a Lebesgue integral, ti Vb {for ti less than 1} is also a Lebesgue integral, earned value of a homogeneous work during any given segment of time is a simple Lebesgue measure. Hence homogeneous works can be directly measured by Lebesgue units completed. This type of measurement is conventionally called Unit Value measurement.

Scaling of resources implies that Vb = Sum of kj c.r [Tfkj - Tskj] for j = 1 to q for q segments such that for fixed quantum works, duration of each segment is reduced to [Tfkj - Tskj] = [Tfj - Tsj]/kj.
The scale factor k and no. of segments q are always 1 for fixed duration works. Fixed duration cannot be speeded up, broken or delayed. If resources are increased in fixed duration works, actual cost increases proportionately.
For fixed effort works, resources operate even in intervals between discontinuous segments of work. Total cost is hence defined for resources over period [Tfq - Ts1] that is start of first segment to end of last segment. Hence, during a break or delay in such works, no incremental value is achieved although cost increases at the same rate as k.

Homogeneous works are amenable to PDCA analysis provided each work package in the WBS are mutually exclusive – that is one work is independent of another in terms of cost, productivity, scale and time. In such cases, PDCA analysis can be carried out as follows:

1. All works are defined as mutually exclusive homogeneous works. The logical relationship between activities is established using conventional Gantt chart and CPM analysis.
2. Dividing the project duration into fixed periods, the planned value curve can be established as cumulative summation of cost functions on Gantt chart over each fixed period. Floats and statistics on CPM analysis may be incorporated in planned values to obtain statistical region of planned value.
3. At any time during the project, a desired planned value of work is scheduled for a control duration. The resources required for this objective may be calculated and sub-optimised by inverting the cost function to find a suitable scale factor k. (Sub-optimization procedure is to be elaborated eventually.)
4. Project managers advise contractors and vendors to supply material, equipment and labour in accordance with the sub-optimized value of k during the control duration.
5. The earned value of work during a period may be different due to actual variations from planned works. The earned value shall be measured at generic instances of time using the Lebesgue measurement function or tracking function. Definition of a simple tracking function is necessary to operate this stage of management. This shall be discussed eventually. The actual resources employed in the duration can also be tracked in terms of material receipts, equipment time cards and muster roll registers.
6. Deviation between generic earned value and planned value designates the difference between slope of the cumulative planned and earned value curves. Whereas, cumulative planned and earned value curves are plotted continuously to present the conventional earned value diagram, the necessary adjustment of generic slope of earned value curve can be applied on the next cycle of management control. If the rectification slopes are within the planned value ranges, attempt can be made for rectification. Otherwise, baselines are to be revised rationally for irrecoverable damages done.

The foregoing analysis is possible only for homogeneous works because the all mathematical calculations viz. integration, differentiation and inversion are simplified on such works. We shall henceforth discuss

• Integration procedure for summation of cost functions into planned value diagrams
• Differentiation procedure for generic measurement of earned value using tracking functions
• Inversion procedure for sub-optimization of scale factor k for desired planned value in scheduled durations
• Statistical definitions for mathematical expectation of planned value in CPM analysis
• Identification of works that are not inherently homogeneous and methods of homogenizing such works