Non-homogeneous Works in WBS for Earned Value Tracking in Projects

Assuming homogeneity of work, introduces many (over-)simplifications in definition of work, as follows.

• Contribution vector c (t) = c is constant during the duration of work; productivity of resources in a prescribed geographical and technological condition is fixed. This means that work is well learnt and resources are sufficiently skilled. Hence, homogeneity may not be applicable to works that follow non-linear learning curves.
• Cost of resources r(t) = r is constant during the duration of work. In reality, cost of resources varies considerably on long range and fluctuates marginally on short range. These cost variations are at best statistical and non-analytic. Hence, it is customary to consider the most probable value of costs as constant in time during the project subject to revision in long range. Similarly, cost reductions for hiring equipments on long term, depreciation of materials over time etc fall outside the applicability of homogeneity.
• Value of work done in a control period Dt is the linear sum of cost of resource contributions to the work. Cost of resource r(c, t) = r is not influenced by the quantity of the resource used and/or another resource. That is cost of resources are mutually exclusive. However, this simple conception is not applicable in over-time and night shifts. Rate per unit weight of steel is dependant on bar diameters and rate per unit area of flooring tiles is dependant on tile size.
• Value of work in a control period is assumed linear with respect to time. However, there are works whose values are not linear functions of time. For example, cleaning the site throughout the duration of the project when defined as a single work, it would be a repetitive cyclic work with several peaks and troughs during the entire project period. Thus, in order to scale the resources, different scales may have to be used at different stages of the project.
• Homogeneous cost function is continuous. Hence, works with earned values associated with distinct points of time are not homogeneous. For instance, work defined with values at delivery of materials (7th day, 20% value), assembly (14th day, 60% value), installation (17th day, 15%) and testing (20th day, 5%) is a work with discontinuous valuation. The cost (or value) function in this instance is V = 0% in t[0,7), 20% in t [7,14), 80% in t[14,17), 95% in t[17,20), 100% in t>20. These cases called incremental works may be associated with same or distinct sets of resources at points of measurement and resources do not appear on cost function. Definition of work for measurement of earned value in this manner is conventionally called incremental method of measurement.
• Since cost function is linear, rate of change of work value per unit time is constant. Homogeneity is not applicable in stepwise-linear works, with different rates in different intervals so that value may be defined as DV = cz.rz Dt where tz less than t less than tz+1. Consider that one set of resources may work during say 30% of work duration to create 60% value. Another set of resources may contribute in a particular proportion for 40% of duration to achieve 30% value and at different proportion for 30% of time to create 10% value. In many cases, the durations in which different resources work may also overlap. Stepwise-linear works are conventionally measured by combined system of measurement, where unit value measurement with respective coefficients is used between each incremental stage.
• When a work is expressed as a combination of two or more cost functions that occurs simultaneously or in close succession, the combined cost function is not scalable. On a control period Dt € U (Dtz for z = 1 to Z segments of time), if cost function is defined as DV = S cz.rz Dtz for z = 1 to Z, the work itself becomes a mini-project. For example, column concrete involves fixing of reinforcement, marking column shoes, shuttering of column sides and concreting of columns. Each activity requires different set of resources and are measured in different units. Some of these activities may overlap in execution whereas others may follow in close succession. These works shall be called composite works.
• Limitations of work front availability may constrain scaling of resources in an erstwhile homogeneous work. For example, 250 piles cannot be cast in one day using 250 pile rig sets and corresponding labour due to operational space of each rig. Mathematically, cost function DVz is bound by Vbz less than Vb for z=1 to Z groups of spatial fronts. This makes mathematical calculations on cost functions difficult until works are split based on front availability. Such works may be called split works.
• Some resources contributing to work may itself be result of other works, say for instance in assembled products such as trusses, gates, steel fabrication, doors and windows. These works require a matrix definition of resource cost as Dr = c’.r’ Dt. The resource work and real work are both separately homogeneous, but are not homogeneous when combined. When the set of intervals in which resource work is performed overlaps with set of intervals in which real work is performed, the cost function may also become non-linear because, resource cost itself is a function of time.
  

In all aforesaid non-homogeneous functions, one can visualize typical deviations of cost function from the format DV = c.r Dt. In incremental works, c.r is directly defined as a Lebesgue measure, whereas in stepwise linear works, c.r is variable over intervals of time. In composite works, there are numerous functions of the linear c.r form for different components in the summation. In split works, application of resource scales are discontinuous on cost functions and in assembled works, cost functions are functions of other cost functions.

In this context, it is necessary to define a procedure that consumes prescribed quantities of a prescribed set of materials, equipments and labour uniformly and continuously in a prescribed time-period. Such procedure shall be called an activity such that activities always have a simplified cost function DV = c.r Dt. Works are linear/non-linear, continuous/discontinuous, simultaneous/overlapping/disjoint, uniform/non-uniform arithmetic/functional combinations of activities.

The major conceptual difference between activity and work is such that physically works are the entities that are managerially reported, although activities are the entities with homogeneous cost functions. Managerial reports of work values in PDCA-EVM analysis are of five types – Scheduling, Execution, Tracking, Measurement and Payment.

We shall now see how this concept can be applied to simplify analysis of each of the exceptional cases of work, so that they remain amenable to PDCA analysis by earned value tracking.