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.