Incremental Milestone Works in Earned Value Analysis

Works with earned values associated with distinct points of time have cost functions of the form

V = Vz at tz less than t less than tz+1

are called Incremental works. The TCM Framework of AACE calls this type of measurement as Incremental Milestone. Definition of incremental works is not amenable to PDCA analysis because resources do not appear on cost function. Cost function is identical to measurement function and hence the advantage of measuring work from two directions viz. deliverables as well as performable is lost.

For PDCA analysis, incremental works need to be homogenized by defining distinct homogeneous work packages with constant cost functions and extremely short fixed durations at instances of increment. The interval between increments is covered by trivial works that cannot be measured or paid.

An example of increment work previously discussed was installation of an equipment. Work was defined with values as –

• Delivery of materials (7th day, 20% value)

• Assembly (14th day, 60% value)

• Installation (17th day, 15%)

• Testing (20th day, 5%)

The cost (or value) function in this instance is

V = 0% at 0 less than t less than 7, 20% at 7 less than t less than 14, 80% at 14 less than t less than 17, 95% at 17 less than t less than 20, 100% at t greater than 20.

In the above example, supply of materials is a work of duration one day and value 20%, scheduled on 7th day. Similarly, assembly is a work of duration 1 day, value 60% and zero cost. This work is scheduled on 14th day. A work called assembly-in-progress may be defined as a trivial work with zero value but non-zero cost function scheduled from 7th to 14th day. If the trivial work is homogeneous, percentage work completed on each day can be tracked and rectification applied whenever necessary. Since, trivial works have zero values; it cannot be measured and paid. From the contractors’ perspective, trivial works have non-zero cost function, to keep track of resources needed to complete work and control over cost of such resources. Sum of the work package at increment and trivial work in between equates the cost and value functions without altering the mode and method of measurement.

Fabrication and erection of steel buildings usually falls under incremental milestone works. Fabrication of a girder beam from a rolled section may involve:

• Receipt of rolled section at yard

• Straightening of section

• Cutting to length

• Grinding edges

• Jointing to length

• Drilling holes to catch mating members

• Placement of end plates and reinforcement plates

• Positioning of intermediate stiffeners

• Positioning of cleats

• Continuous welding,

• Providing priming coat

• Pre-erection painting.

Each of these tasks is in-turn distinct activity consuming different resources at different proportions and associated with different costs. However, value is attached generally only at the beginning and end, say 40% at receipt of materials and 25% upon completion of fabrication.

If a managerial interference is extended on the scheduled day of completion of the fabrication, it is likely that project delays and cost over run might remain unnoticed for corrective action. Hence, a suitable reporting system is needed to track the progress of this fabrication.

In order to homogenize the fabrication work, trivial works are defined for straightening, jointing, drilling, welding, painting etc in WBS. During detailed planning, each of these trivial works also should be scheduled assigning resources. The number of beams completing each stage can be reported as tracking parameter every day. Thus, completion of works can be compared against performance of resources in PDCA analysis and corrective measures can be taken whenever these are warranted.

Trivial works are not associated with physical measures of completion. For example, there is no percent value complete associated with fixing of end plates on the girder. That is, trivial works accrue cost but do not accrue value. During physical measurement and payment, trivial works shall not be measured or paid, unless all the trivial works leading to the incremental work is completed.

Incremental works also justify the necessity of separating planned value curves from planned cost curves. Although, value is accrued at completion of work, cost is continuously accrued during the work. Comparison of earned value against planned value is of least administrative significance in this case, since it does not enlighten about delay or cost increase during the work. On the other hand, comparison of actual cost against planned cost provides insight about these deviations.

Definition of incremental works is common in software development projects, design projects and film industry because intermediate deliverables do not possess stand-alone value. In software industry and in design projects, person-hours are the most significant resources. Planned value and cost is defined in terms of the person-hours. Earned value is defined by the completion of work on hand. It is hence necessary to define intermediate deliverables to be tracked albeit without value, so that corrective actions may be taken when such deliverables do not meet targets. After final delivery, the work is valued by physical measurement and earned value curve is updated.

If the accrual of value is associated only with financial transactions such as billing to client or payment to vendors, trivial works can be associated with earned value and flagged off for payment at later stages of work.

Staggering of financial transactions can be seen in many incremental works. For instance, in the steel building example, client is unlikely to be billed or contractor to be paid for completion of fabrication of every girder. Such transactions may occur for instance after completion of say fabrication of columns, braces, purlins, runners, base plates etc that is after entire lot of fabrication is completed. In such cases, it needs to be specified that transactions are staggered until another incremental work is completed.

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.

Tracking Function for Earned Value in Construction

PDCA analysis is extremely simplified for homogeneous works, if measurement of work can be simplified in generic instances of time. It has been shown that for homogeneous works, measurement of work during ith segment of time Vi = ti Vb {for ti<1}. This can also be extended to show the validity of superposition in work measurement as : Vi+j = ti Vb + tj Vb = Vi + Vj. Hence, measurement functions are linear and due to the absence of intercept, should also be scalable.

This is possible if and only if measurement function Vb = g (x1, x2, x3,… xm) is homogeneous in at least one variable x1, so that g (zj x1, x2, x3,…xm) = zj g (x1, x2, x3,…xm). Thus, the measurement function can be rewritten as Vb = x1. x2…xq j (xq+1,…xm) and is homogeneous in the product function x1. x2…xq. This product function can serve as a quick measure of work at generic instants of time, since it does not involve measurement of all Lebesgue integrals. We shall call this function as a tracking function, q = x1. x2…xq. Therefore, measurement function is Vb = q j (xq+1,…xm) such that zj Vb = zj q j (xq+1,…xm).

For instance, a common measurement function is Vb = NLBDR (N=numbers, L=length, B=breadth, D=depth and R=cost of unit volume). For homogeneous works, there always exist tracking functions in the set of functions {N, NL, NLB, NLBD} which can be used as a quick measure for work during generic intervals of time. Thus for footing concrete, no. of footings may be a tracking function. Similarly, total length of road may be used as tracking function for pavement carpet, total area for plastering and percentage of total volume itself in mass concrete works.

There is a tendency to assume the measurement function itself or percentage thereof as the tracking function. Although, there is no technical error in this assumption, it warrants complete measurement of works in all short intervals of time. It is necessary to keep the tracking function simple in measure albeit under the expense of a complex definition of work package. For instance, conventional measurement of brick masonry in walls by short wall-long wall survey minus deductions is not viable to simple tracking. Length of wall can be used as a tracking function instead of total volume, if wall masonry is defined at instances – below-sill, sill-lintel and above-lintel units with respective deductions applied. The difference is basically that all site quantities are pre-measured and correlated to a unique tracking measure.

Not all measurement functions fall in the category of NLBD measurement. For volume of truncated pyramids in footings, the measurement function is not a product function, which is why the non-tracking component is generalized above as j.

Trivially, the measurement function may also be defined as Vb = S qj j (xq+1,…xm), such that measurement during ith segment of time is the sum of measurements in j works of similar nature. The measurement function here is indeed a summation function. Similarity of works is such that other functional components except tracking function remains constant for all terms of summation. When a work is defined as Vb = q j (xq,…xm), we call the work as a simple work. When the work is a summation of simple works Vb = S qj j (xq,…xm) we call it compound work. In NLBD measurement function for brick masonry in walls, the function may be (N1L1BD + N2L2BD + N3L3BD) for instance, in which case this will be a compound work.

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
  

Formulation of PDCA Cycle in Construction

We shall tentatively define a mathematical formulation for value of a work package with complementary resource contributions and permitting necessary calculations as follows. Value DV of a work package in a control period Dt is mathematically expressible as a cost function (Read D as delta):

DV = f (c,r,t).Dt

Where c is the contribution (vector) of resource efforts in unit time to achieve unit quantum of work, r is the cost (vector) of resources and f is a function of productivity, cost and time. Resource contribution and cost are defined as vectors because they apply to set of relevant resources. The cost function has following characteristics:

1. The cost function can be integrated or summed up over duration of work. The sum of cost function over duration of work is the budgeted cost of work or total planned value of the work.


2. The cost function DV need not be a continuous function in time. Integration or summation over durations is possible even on discontinuous functions. However, sum or integral V of cost function over duration T should necessarily be continuous, such that rate of change of work value in real time and real cost may be calculated. As an extension, summation of integrated values of different works during a given interval of time need also be continuous.


3. The cost function DV can be scaled for resources, costs or time by a scalar as f (k1c, k2r, k3t)Dt to scale the value of work in the control period Dt. Scale cannot not be a vector, because different resources cannot be scaled differently to achieve a scaled value of work due to the complementarities of resources.


4. The cost function DV should be invertible because during planning, it is necessary to determine scaled contribution k1c for given quantum of work in given control period.


5. Cost functions DV are monotonously increasing functions (straight upward lines, exponential curves, S-curves etc) because work cannot reduce with increasing effort or increasing time. This means that cost function does not have a zero intercept value at any time except if necessary at start of work.


6. Cost functions DV are always positive. That is no resource is consumed negatively and no resource is with negative cost.

The cost function DV is a bounded function that is not defined in - ∞ to + ∞ or being positive function in 0 to + ∞. Value of work executed in control period is finite and sum of cost functions are limited to total planned value of work. The bound of cost function may be applied in three forms:

1. Fixed quantum work is such that duration of work may be reduced by uniform scaling of resource such that the work value remains constant. Most works in construction projects fall in this category. A delay of work reflects as a flattening of slope of earned value curve. Delay of work does not necessarily cause a cost increase. Slope of cost curves follow the value curves unless cost or wastages cause escalation. The bound of cost function is fixed and scale of resources and work duration are variable for given work value according to the inverse of the cost function.


2. Fixed duration work is such that duration of work cannot be altered by scaling resources, such as in curing of concrete. Fixed duration works cannot be delayed and earned value curve always follow planned value curve. Scaling of resources in fixed duration works directly results in cost over-run, that is cost curves are steeper if resources are wasted on fixed duration works. Bound of cost function, duration of work and scale of resoources are all fixed in this work, and cost function is a continuous function of time, that is there cannot be a break in execution of work.


3. Fixed effort work is such that a limited effort of resources contributes into the work. Plant-in-site, technical supervision and timbering of excavation is examples of fixed effort works. Delay of fixed effort work is always caused by delay in other (fixed-quantum) works. When physical time of resources in fixed effort works is not utilized any delay invariably causes cost-over-run. Delays present themselves as flat earned value curves and actual costs become steeper at all delay. The bound of cost function and resource efforts are fixed in these works. The cost function may not be continuous in time, so that net duration of work may be longer than the duration according to inverse of cost function.

In the PDCA cycle, the cost function enables to define planned values as a function of resources employed, their anticipated costs and the duration in which they are employed. Each work package on the WBS defines or quantifies a certain bound Vb for work value based on fixed quantum, duration or effort criterion. Summation of the values of works on scheduled period provides the planned value curve.

The binding value of the work package is determined by measurement of deliverable that necessitates a separate method of measuring completed work. For this purpose, we will define a measurement function:

Vb = g (x1, x2, x3,………….. xm)

Where x1, x2, x3,………….. xm are vectors of Lebesgue measures such as lengths, areas, volumes, weights and counts of deliverables in the work package. Generally, on all projects measurement of deliverable as a Lebesgue integral is possible after completion of work. However, for application of EVM-PDCA cycle, it is necessary to define a Lebesgue measure such that deliverables can be measured during execution and before completion of work. We shall call such a measure as a tracking function.

The resource efforts required during a control period to achieve a certain value of tracking function can be calculated since cost functions are invertible. For all Lebesgue measurement of works, albeit after completion by measurement function or during execution by tracking function, the earned value of project can be calculated at generic time. The work performance can be controlled during a control period by scaling the resources, so that slope of earned value curves may be varied at will.

Thus, the cost function and measurement or tracking function is useful for control during PDCA cycle, since it enables separate measurement of all quantities DV, k, c, r, Dt and deviation of any measurement from plan can be immediately acted upon for managerial control.

Earned Value Method from Management Control Perspective

Construction Management Part IV

From a management control perspective, an analytical model of a system should also be capable of offering solutions to problems. Hence, it is necessary that PDCA analysis using Earned Value Method, provide insight on the corrective action for variances during assessment phase. Thus, it is necessary to address the following aspects of earned value representations:

• Permanence of project delays or cost over-run: (semi-)objective indications are needed to assess how far of a deviation is an irrecoverable damage. Attempting to recover a permanent delay or expenditure is meaningless or sometimes harmful.
• Causes of over-expenditure: Extent to which planning errors, price escalations, sub-optimal resource allocation, wastage of resources or idling of resources during delays contribute to cost over-run.
• Causes of delay: Extent to which planning errors, low productivity, insufficiency of resources and uncoordinated resources contribute to schedule delays

Heuristically, in addition to measures of variances in the project as a whole, additional measures of control are necessary to address control-related issues as contained above. Control is essentially an attempt to change the slope of earned value and actual cost curves at will, hence these measures need to be based on the slope or derivative of the PV,EV and AC curves at generic instants of time and generic stages of value of the project.

To elaborate, consider that a smeared PV region is created probabilistically based on optimism of planning. The region would also accommodate deterministic early start (ASAP) or late start (ALAP) PV curves. For different cash-flow conditions, smeared planned cost PC region may also be created with deterministic ASAP/ALAP planned cost curves. (PC curves may become probabilistic themselves, if resource costs are probabilistic). When earned values and actual costs are recorded in this system, it may also be possible to plot the costs as required to earn the value by ASAP/ALAP schedules – we may call this as earned cost EC region. (It is not emphasized that planned costs and earned cost would be identical to planned value and earned value respectively if Just-in-time inventory control is applied).

This approach can provide impressive conclusions, such as; slope of optimistic PV at generic value may be upper bound of expediting project by resource enhancement. Slope of ASAP/ALAP PC may indicate upper bound of cash flow requirements. In a well-controlled project during a delay, EV follows PV at least pessimistically, slope of EV tends towards optimistic PV and AC is comparable to EC.

Projection of cost over-run and delay also fares well when slopes are accommodated in the forecast, because deliberate control attempts to compensate delays by resource adjustments and cost increase by resource performance. As a result, during control the slope of EV curves change towards slopes of optimistic and early-start PV curves. Thus, affine translation of suitable PV curve and suitable affine rotation for recovery is a realistic transformation than simple affine scaling for forecasting. In this analysis, rotations are limited by the slope of optimistic PV curves, so that any delays that cannot be rectified by following optimistic PV curves remain irrecoverable. Delays that are within the difference between ASAP and ALAP may be recovered without cost-over-run.

For a given EV curve and corresponding EC curve, deviation of actual cost from absolute maximum earned cost may represent an irrecoverable expenditure. Deviations between ASAP and ALAP earned costs may be recoverable by better control.

[All foregoing discussions are strictly only mathematical hypothesis and do not amount to inference. Actual behaviour of earned value curves in reality are subject to multitude of parameters. Inference is sought to be developed after actual implementation of earned values on different projects.]

Testing of aforesaid methods, leave alone their practical application, is formidable due to the extensive data capturing and mathematical computations involved. A suitable methodology needs to be devised for this purpose in order to derive fruitful results. Notwithstanding this difficulty, it stands that the slopes in EVM are as significant as the values and indices both from control perspective and forecast analysis, where recovery curves are used at generic instances.

Analysis by Earned Value Method

(Construction Management Part III)

Quantity of work planned or completed in specified duration should be measured with a reliable variable. The measure should be applicable with same units for all kinds of work and should be relatable to the quantity of resource-effort and duration of work.

Budgeted cost of work BCW is a reliable measure in planning (BCWP) and tracking (BCWC). This measure can be made dimensionless by considering the ratio over budgeted cost of the project. For purposes of analysis, fraction of budgeted cost of project in works (or parts thereof) planned during a control period is called the Planned Value in the control period. Fraction of budgeted cost of project in works (or parts thereof) completed during a control period is called the Earned Value in the control period. Tracking projects by earned value is called Earned Value method and is amenable to PDCA analysis.

A PDCA analysis following above approach attempts to quantify problems in the project as deviations in planned value, earned value and actual cost over project duration elapsed. Such measures of deviation that require corrective action can be identified as:

• Resource variance RV when Actual cost to date is greater than Planned Value to date.
• Schedule variance SV when Earned value to date is less than Planned value to date.
• Cost variance CV when Actual cost to date is greater than Earned value to date.
• Time variance TV when Planned duration for current earned value is less than duration elapsed.

These definitions of variances measure non-concurrence of projects to plan. Many of these variance measures may appear redundant and much literature has examined the efficiency of these measures under various situations. A conceptual description of earned value method may be presented here without emphasizing the measures of variances.

Earned value analysis identifies two major contingencies in a project – delay and cost over-run. Delays are usually caused by insufficient or inefficient resources; cost over-runs are caused by purchase price escalations and wastages. Delays might themselves cause cost escalation during idling of resources. Moreover, delays and cost rise might be planning errors, as well. On the EVM curve, a downward deviation of earned value (EV) curve from planned value (PV) indicates a project delay. An upward deviation of actual cost (AC) from planned value indicates a cost over-run.

For a delay, an affine transformation of PV curve may be used to estimate the projected project completion time. It is conventional to scale the PV curve to arrive at a proportionate delay at project closure. The scale factor may be ratio of PV/EV at generic time or Actual Time (AT)/ Scheduled Time (ST) at generic EV. Similarly, an affine transformation by scaling PV curve is used to estimate the cost at completion, the scale being the ratio AC/EV at generic time.

These estimates do not deterministically predict actual delay or cost over-run because the PV curve is itself not deterministic. Indeterminacy of PV curves may be due to various reasons:

• Probabilistic nature of planning: Optimistic PV curves are steeper, accommodates minimum float and does not allow for schedule compression. Pessimistic PV curves are flatter and allows for expedition during execution.
• Adjustment of float in non-critical tasks: Early start PV curves are steeper during initial stages and flatter at later stages. Late start PV curves are flatter at initial stages and steeper at later stages.
• Inventory management method: By definition, PV curves are plotted with JIT inventory management basis. When materials are purchased in bulk and stored in central stores, planned costs (PC) are likely to be greater than planned values during initial stages of the project. In such cases, it may be necessary to plot PC as a separate curve in the time domain.

Although, PV curves are in reality smeared over the time domain by probabilistic nature, adjustment of float and cash flow parameters, it is customary in EVM analysis to regard PV curve as a deterministic curve. Thus, estimates of project delays or expenditure over budget at completion turn indicative and non-deterministic.

Summary of PDCA Analysis by Earned Value Method

(Construction Project Management - Part V)

The essence of PDCA analysis using EVM method in project management may be summarized in following procedures:

• Probabilistic planning and scheduling; delineation of planned value region
• Calculation and depiction of early start and late start schedules
• If inventory is not Just-in-Time, probabilistic or deterministic inventory flow analysis to calculate planned costs
• Continuous measurement of earned value and actual cost, such that curves are differentiable and slopes can be calculated
• Superposition of inventory flow on earned value to identify earned costs
• Comparison of earned values to planned values; (for no deviations thereof,) actual cost to planned cost; (and for deviations thereof,) actual cost to earned cost
• Formulation of delay or cost recovery curves by proposing suitable recovery slopes
• Analysis of proposed recovery curves for inequalities defined by smeared planned value region, float adjustment and cash flow constraints
• Forecast of projected delay and cost over-run based on approved recovery curves

Intensive application of EVM analysis requires mathematical formulation of work values and costs that are amenable to calculations at all stages of analysis. Value is associated with the numerous works in the project and cost is associated with the numerous resources that contribute to various works. Accordingly, each work package in the WBS should allow following mathematical operations:

• Scheduling for a planned value of work in a given duration; calculation of effort of each resource required for this planned value.
• Execution of certain earned value of work over a control period mutually exclusive of other work packages in proportion to the actual effort of resources employed during the control period.
• Measurement of actual resource efforts in the control period and earned value of executed work.
• Assessment of deviation between planned value and earned value over the control period and attribution of the deviation to deviation in resource effort.

The resources that contribute to a work package in foregoing formulations are always complementary. No resource can independently affect quantum of work. Resources jointly act in functional combinations to produce a quantum of work. Hence, value of a work in unit time depends on the cost of the set of relevant resources and effort of each resource contributing to produce unit work in unit time. When resource supply is enhanced during a work with an intention to expedite a work, such enhancements has to be uniform on all resources in accordance with the functional combination of resources in the work. Increasing one resource without proportionately increasing others would not result in meaningful transactions vis-à-vis increase of work performed and/or reduction of work duration.

In economic terms, two or more resources are said to be complementary if they are used together during consumption. In construction, most resources are complementary in specific contribution ratios in activities. Three types of complementary resources can be identified in economics (of construction):

1. Perfect Complements: Bolts-nuts and Machines-operators are complementary resources in all instances with fixed mutual contributions. For many practical purposes, two or more perfect complements may be consolidated as a single composited resource. However, during definition perfect complements may have distinct units and distinct values.
2. Imperfect Complements: Majority of construction resources fall into this category, because complements combine in different non-fixed ratios in different consumption events. Relative proportions of resources may vary in reality and consumption occurs with non-fixed contributions.
3. Non-mutual Complements: This type of complementarities occur when consumption of one resource requires a complementary contribution of another, but consumption of second resource does not require a contribution of first. For instance, when a process consumes a labour day, tool-days are also simultaneously consumed. A labourer may use different tools on different days; hence, consumption of a particular tool-day does not necessarily indicate a labour day.

Project Management - A suitable system?

The issues in foregoing post can be offset by replacing the conventional linear incremental system by cyclical systems such as Deming-Shewhart cycles. Deming-Shewhart cycle of Plan-Do-Check-Assess described in 1930s is conducive to project management. It is also referred to in PMBOK of Project Management Institute and TCM framework of Institute of Cost Engineers.

The cycle attempts to plan projects in measureable parts, execute them, measure them and take corrective action. Frequency of planning, execution, measurement and correction vary among projects. Thus, the PDCA cycle creates a short-lived system and closes itself at predetermined intervals. Linear incremental systems on the other hand are enduring and intermittently improved at unforeseen contingencies.

Breakdown of project into measureable packages of work is imperative to project management because PDCA management is equivalent to measurement. An enumeration of works on a project is called a Work Breakdown Structure WBS. Works may be measured in three methods:

1. Quantity of work executed and quantity of resource-effort in fixed duration of time.
2. Quantity of work executed and duration consumed for fixed efforts of resources.
3. Duration consumed and quantity of resources at fixed stages of work.

The second model of measurement is generally used in pareto-optimization and operations research. It makes the domain of analysis extremely multi-variate. Resource-efforts seldom remain constant or cyclic in practical projects; hence, this model is not amenable to cyclic system. Controls based on this model constitute Go/No-Go Decisions or Binary Control techniques, wherein trial runs about a method can be used to decide about application of the method to an ensemble.

The third model of measurement is suitable in Post-Control or Retrospective Control techniques. It is by far the most accurate model of measurement and hence followed for commercial operation of projects – to receive and make payments. However, since works are not repetitive, this method is not suitable for implementation of cyclic systems.

By far, the most suitable method is to analyse work done and resources employed in fixed spans of time because time domain is uni-dimensional. All variables can be related to time and cycle frequency can be easily defined in time. Monitoring work on limited periods offers possibility of imposing Steering Control. This has been used in defintion of Meredith-Mantel Cybernetics (2006).

To conclude, it is necessary to measure work for fixed effort before the project (or part thereof), for fixed duration during the project (or part thereof) and for fixed work after the project (or part thereof).

Project Management - Difficulties

An organization is a set of resources ordered by a mission driven towards a set of predefined objectives through a set of operations. Management is the profession of directing resources towards objectives through pragmatic decision-making.

Management seeks to impose uniformity of decisions and collective binding on the decisions across the organization, because output of organization is collective effort of different individuals. The most accepted method of management in this regard is the systems approach of management. This approach attempts to formulate a systematic procedure for a majority of routine works in the organization. Contingencies are situations that are not covered by the system, and discretion during contingency updates the existing system.

Operations are of various types. It may be continuous as in line production, intermittent but repetitive in batch production. An operation that has a distinct life span is called a project. A project that transforms the quality of geographical space is called a construction project.

The basic distinction of Project management is that mission, objectives, organization and operations of a project are transient. This throws many challenges on project management:

1. Definition of projects and their objectives are one-time events. Hence, they tend to be incomplete (causes scope-creep), optimistic (causes over-run), non-comprehensive (causing conflict in various operations within the project) and ineffectively communicated (causing conflict among team). When projects are technical, as in construction, they also tend to be complex and procrastinated for detailed technical designs.
2. Processes in a project are short lived; documentation & data-capture are non-routine. Identification of suitable system, training project personnel on the system, real-time implementation and adaptation for contingencies – all may be individually or collectively longer than process duration. Hence, project personnel tend to work without a proper management system; projects (or processes thereof) close with unsatisfactory completion or gross failure. In construction projects, projects are subdivided into activities that are even more short-lived than the project and each activity may need a technical system for proper control.
3. Project organizations are ad-hoc. They are assembled specifically for the project and are hence subject to considerable inertia. Introduction of a project is itself a perturbation on existing system. In construction projects, project organization involves different firms with different objectives. Ad-hoc assembly of different organizations are governed by documented contracts. Limitations in project definition and system usually make contract documents non-transparent and ambiguous. Technical assessment and/or value engineering by contractors at pre-tendering stages is skipped.

Some Observations on Project Scheduling

Many progressive engineers have attempted to plan and schedule moderate construction projects mostly unsuccessfully. This is despite the most advanced academic theory, sophisticated computational equipments and rigorous automated procedures of modern times. Successful project management never emerges despite earnest efforts: is this due to differences between theory and practice, inefficiency of modern technology or mismanagement of scheduling engineers?

Whatever be the reasons, scheduling has remained a mystery that demoralizes engineering community. Engineers with unsuccessful experience in project planning have given up the idea of rigorous scheduling as unpractical and have adopted management by day-to-day control of activities without a pre-engineered baseline. If a schedule is not useful in practice, the effort used in preparing it is an unnecessary overhead – cost can be saved by not scheduling, rather. Many other engineers still prepare schedules, for documentation purposes alone with little managerial applications. Few construction managers make contractors prepare weekly or monthly schedules and track project progress based on the submitted schedules, without attempting to revise or refine the plan holistically. Control is applied on what is, but thought is not applied on what could have been. Many construction companies also feel that conventional and run-of-the-mill projects such as construction of apartments, does not derive any advantage by scheduling, even if scheduling as a method were successful.

However, it cannot escape our attention that most projects around us (including conventional and run-of-the-mill types) have not been completed on time. Given the competitive economy of the day, builders are required to give consumers more utility at lesser costs. The importance of cost reduction on intangible components such as time cannot be over-emphasized, but to date there is no effective mechanism to make at least 80% control on construction projects possible.

Certain observations can be made about current practices of scheduling

Rigorous Schedules

1. Engineers tend to plan and schedule the project at initial stages rigorously and submit baselines for on-site control. This requires larger initial effort and more important, rigorous schedules require equally rigorous implementation and monitoring leading to failure at implementation stages.


2. Rigorous schedules cannot necessarily accommodate potential contingencies and unforeseen problems. There is no simple technique such as allowing contingencies in project cost estimates when it comes to project scheduling.


3. In most cases, information required for rigorous schedule such as extent of working capital available, interest cost of borrowed capital and extent of availability of materials, equipments and labour on credit is uncertain. Managers with sufficiently long experience to quantify the above parameters are seldom well versed with technology of scheduling. The cost of collecting information on a most-probable-basis by surveys is usually very high.

Shortest-path Schedules and Working Capital Management

1. Planning engineers usually concentrate on minimizing the time period of construction while scheduling. However, shortest period of construction need not be necessarily be the objective of the company for most moderate constructions. Expedient construction is important for contracting organizations executing time-bound projects; for other companies faster construction may involve escalation of costs, in terms of working capital requirements. This makes shortest-time-schedules to fail due to apparent working capital mismanagement.


2. Working capital on projects may comprise of borrowed and relatively free funds, such as own investments, retained earnings, capital released by sale of previous projects or capital generated by sale of developments in advance of construction.


3. Consider a stage of project where company needs to borrow capital in order to operate the project on schedule. If the company expects realization of a free fund in few days, it may attempt to deliberately delay schedule rather than increase liability. Under these circumstances, it is necessary for scheduling engineer to evaluate the cost of delay of project at every stage of construction. If the cost of delay is greater than the cost of borrowed capital, the company needs to borrow and maintain the schedule whereas if it is not company should delay the project.


4. The cost of delay includes interest cost of already borrowed capital during the delay period, cost of hired equipments and daily labour in the middle of work, overhead cost such as site office rents and site engineer salary during the delay period, loss of credit advantage on already purchased materials and hidden cost due to staggering of turn-over. Unfortunately, most minimum-time-schedules evaluate projects for a given time-frame but do not analyze possible cost-overruns due to prescribed delays at every stage of construction. It is necessary to make the schedule an information system for managerial decision making rather than as a document of tracking a hard-and-fast criterion.
5. The objectives of scheduling may be different for different organizations. Scheduling objective may be to

• Reduce time period of construction irrespective of any actual or hidden cost escalation


• Minimization of actual cost which may or may not require reduction of duration of construction


• Minimization of hidden costs which may or may not require elongation of duration of construction

Managerial Coordination

1. Construction assignments are seldom undertaken single-handedly by one person or organization. There are a number of small players involved – architect, engineers, material suppliers, sub-contractors, labour contractors, casual labourers, equipment suppliers. It is important that information regarding the project schedule should reach each concerned organization or person by coordination.


2. If one of the organizations fails to make a commitment there is necessarily a schedule delay usually accompanied by a cost over-run. Procurement schedule of goods or services from different people is seldom created from a project schedule. Consequences of delay in procurement need to be highlighted and suitable alternate arrangements needs to be planned.


3. Sometimes schedules itself affect project organization. Consider for example a linear running meter process such as laying drainage channel. The process is characterized by a continuous fast process say excavation, followed continuous slow process say concreting the channel, then by a fast process say plastering of channel walls. Suppose there is also an imposed restriction that open excavation should be closed with cover slabs after a limited number of days. The direct approach of scheduling may be to consider the project as four continuous events: excavation starting from the first day, concreting starting from the second day, plastering starting after curing period say seven days followed by placement of cover slabs from tenth day. This schedule may not be successful because excavation and plastering being relatively faster jobs are out of phase with concreting. This may either result in idle labour/equipment for excavation and plastering or employment of plastering and excavation labour/equipment intermittently. Else it may be necessary to perform double work of placing the cover slab after excavation/concreting, removing them for concreting/plastering and replacing them again. The schedule needs to assist the project administrators in deciding which method of scheduling is best.


4. Schedules should also make parametric study on procurement of labour and equipments. For instance on a job of raising n number of RC columns what is the least cost schedule using m column boxes out of which k boxes are owned and (m-k) are hired? The schedule in this case should consider the number of days of job for different values of m, and calculate labour and other costs during that period to select that value of m that gives least cost. Academic routines are available today to treat such types of problems but these are not practiced and time tested.


5. Project plans should also assist the site engineers on day-to-day decision making for tracking purposes. For instance, consider a continuous concreting job with tight schedule of prescribed quantity of concrete per day. Suppose due to some unforeseen reason such as failure of equipment, only 75% of the prescribed quantity could be completed on a day. Is it advisable to carry over the remaining 25% to next days work or to complete the work on the same day by staying over-time? For this purpose, schedules need to specify the opportunity cost of cost components with respect to the work activity and project stages.


6. Many times top management of companies is not capable of appreciating schedule information in paper. Planning engineers may not possess vested powers for implementation of the schedule. Under such conditions, engineers may have to simulate different possible schedules in order to objectively make advantages of adhering to the schedule evident.

Some Guidelines

Based on the above observations, some guidelines can be arrived to make practicable work schedules.

Schedules should be on multiple tier architecture in following levels and characteristics.

1. Each activity such as concreting involving shuttering, bar-bending, placing of reinforcement, concreting and curing should be worked out for different parameters at one level to form a database of different work organization such as item rate, daily labour, lump sum, RMC, hired/owned equipments & labour contract for activities. Opportunity cost of parameters such as delay, credit, cost of capital etc need to be quantified for each activity. This database then becomes reusable again and again on projects.


2. The second level of schedule may consist of phases of the project such as foundation, floor concreting, painting etc. To begin with, the minimum time schedule may be applied on these project phases. The procurement and cash flow schedules may be derived from this shortest-time schedule. At this level on a generic day, various activities take place. Opportunity cost of parameters such as delay, credit, cost of capital, overhead expenses etc at every stage may be quantified by adding those for all activities at that stage.


3. The third level of schedule comprises of different project schedules incorporating parametric differences at second level. With broad outlines specified on this schedule it may be possible to exercise appropriate control over the schedules prepared by sub-contractors on periodic basis.

This multiple-tier scheduling may be academically similar to superimposition of optimization by dynamic programming with optimization by network programming.

The schedules prepared in the above format may be operated in the following manner

1. For every project only the third level project planning needs to be done because database of first level and arithmetic logic of second level scheduling may be available or may be developed based upon actual requirements. Thus, schedule becomes an information system for managerial decision making.


2. Let working schedules be prepared by subcontractors on periodic basis. The working schedules may be studied and modified against fair schedules that represent the requirement of the projects.


3. Monitor the schedules for project objectives and wherever there is a deviation, its impact needs to be quickly analyzed and corrective measures may be taken. If the deviations are due to unpredicted practical conditions the logic of schedule needs to be revised.


4. It may be better to employ singular person within the organization or external agency to be in charge of entire project, including interaction with auditors and financiers.


5. Unlike a building plan or structural detail, project schedule is not a document to be followed at a project. It is a language to interact. Schedules are similar to schemes prepared by architects. Schemes change many times during discussions between architects and clients; the changes diminish as design and construction are progressing. Schedules need to change during construction; all associated features need to be reanalyzed.

The aforesaid methodology is nevertheless, not ruthlessly practiced or time tested in the knowledge of the author. Suitable modifications or grossly different methodologies to confront the problems may also be possible. The objective of this document is to create awareness among construction companies that academic research on schedules is better sponsored and carried out by the industry with association from academia and software development companies