5.5. Equation Formulations in Workforce Management
We begin with the workforce equation, which is formulated as an asset stock accumulation. In Figure 5.21, the workforce at time t is equal to the workforce at time t−1 plus the difference between the hiring rate and the departure rate over the interval dt. The initial workforce is 200 workers, deliberately chosen so that the factory starts in a perfect supply-demand equilibrium with production equal to shipments of 1 000 refrigerators per month. This equilibrium number of workers is obtained by dividing demand of 1 000 by worker productivity of five refrigerators per worker per week. The hiring rate is the sum of the average departure rate and workforce adjustment, a formulation we will return to later in the equations for hiring.
5.5.1. Departure Rate – Standard Formulation for Stock Depletion
The departure rate depends on the ratio of workforce to normal length of stay. Intuitively, the ratio makes sense. The more workers employed in a factory the more of them are likely to leave in a given period of time simply as a result of normal turnover. On the other hand, the longer the normal length of stay, the fewer workers will leave in a given period. Here we assume that the normal length of stay is 50 weeks, so with an initial workforce of 200 workers the departure rate is 200/50, four workers per week. Notice that the dimensions of the equation balance properly. By dividing the number of workers by the normal length of stay in weeks, the resulting departure rate is correctly expressed in workers per week.
Figure 5.21. Equations for workforce and departure rate
The algebra of departures is a specific example of a standard formulation for stock depletion. In many (though not all) practical situations, the outflow from a stock accumulation is proportional to the size of the stock. A familiar example is the outflow of water from a bathtub, which is proportional to the depth of the remaining water. An industry example is the national or regional scrap rate of cars, which is proportional to the total number of cars on the road. The proportion is determined by the normal lifetime of a car, so the scrap rate is equal to the number of cars divided by the normal lifetime. Stock depletion can of course be more complex and depend on conditions other than the 'local' stock itself.[] Nevertheless simple proportional loss is often a good first approximation.
[] The quit rate of employees (or departures) might, for example, depend on relative pay. So then the standard depletion formulation could be modified as follows:
Departure Rate = Workforce/Length of Stay
Length of Stay = Normal Length of Stay * Effect of Relative Pay
This new effect could be formulated as an increasing function of relative pay that takes a neutral value of one when relative pay itself is one (i.e. when there is pay parity).
5.5.2. Hiring – Standard Formulations for Asset Stock Replacement and Adjustment
Factory hiring (see Figure 5.22) includes the replacement of those workers who leave plus adjustments to the workforce size deemed necessary as a result of planned changes in production. These typical pressures on hiring are captured with standard formulations for asset stock replacement and adjustment. The hiring rate is equal to the average departure rate plus the workforce adjustment. First, we will investigate the effect of departures on hiring. A factory producing at a constant rate needs a stable number of workers, which can be achieved by hiring new workers at the same rate existing workers leave. Replacement is a kind of benchmark or anchor for hiring and the simplest equation would set the hiring rate equal to the departure rate. However, in practice, the departure rate takes some time to measure and factory managers may prefer to wait and see how many workers leave in a given period rather than activate hiring for each individual departure. This need for assessment suggests an information smoothing formulation. Hence, the hiring rate depends in part on the average departure rate, which is the actual departure rate smoothed over a period called 'the time to average departures' – four weeks in this case. The precise formulation is written as: Average Departure Rate = SMTH1 (Departure Rate, Time to Average Departures) {workers/week}. SMTH1 is just a convenient shorthand for the standard information smoothing formulation first introduced in production control (Figure 5.11). This so-called 'built-in' function replicates the stock accumulation process at the heart of smoothing without the need for the modeller to create all the symbols. The function is called SMTH1 because there is just one stock accumulation in the underlying structure. (More complex smoothing and estimation procedures can involve two or three stock accumulations in series. See Appendix 2 for further details on smoothing, and in particular footnote 13 to which the appendix refers).
Figure 5.22. Equations for hiring
The workforce adjustment is defined as the difference between desired workforce and workforce, divided by the 'time to adjust workforce'. This equation is a standard asset stock adjustment formulation (just like the correction for inventory in Figure 5.12) representing purposive goal-seeking behaviour – in this case factory managers' efforts to bring the workforce in line with desired. The time to adjust workforce is set at 8 weeks to represent a cautious approach to hiring consistent with factory employment commitments.
5.5.3. Workforce Planning
The equations for workforce planning are shown in Figure 5.23. Desired workforce is a key formulation, for it is here that factory managers decide how many workers are needed. The main driver is desired production. The more refrigerators the factory plans to produce per week, the more workers it needs. The transformation from desired production into desired workforce, however, is itself a decision-making process, requiring both time and judgement. The formulation recognises the administrative nature of this connection between production control and workforce. There is information smoothing because it takes time for people to monitor the relevant information and reach a decision on desired workforce. Judgement is involved because worker productivity is not necessarily known objectively. Instead there is estimated worker productivity. Currently, the estimate is set at five refrigerators per worker per week, the same as actual productivity, but in principle and in practice the estimated and actual productivity can differ, with important implications for factory performance.[] To summarise, the overall formulation says that desired workforce is the ratio of desired production to estimated worker productivity, smoothed over a period of time called the 'workforce planning delay'. Smoothing is achieved once again by using the SMTH1 function, with the planning delay set at four weeks.
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Consider the effect on factory output and production dynamics if the estimated worker productivity is higher than the actual worker productivity. In other words factory managers think workers are (or should be) capable of producing more refrigerators per week than they actually do. So, for any given desired production, the desired workforce is too low. It is a good exercise to simulate the consequences of this bias using a scenario where the retail order rate is held constant at 1000 units per week, and workforce is initialised at 200 workers (the equilibrium workforce). You can run this thought experiment using the production and workforce model. Open the icon for the 'estimated worker productivity' and insert the value '1' in the step function to create a situation in which estimated productivity rises from 5 to 6 refrigerators per worker per week in week 10 (a 20 per cent overestimate). Also, make sure to set the step change in retail order rate to zero in order to create a steady demand pattern. It turns out that when estimated productivity rises, the production rate falls at first as expected. Then it recovers and even overshoots the retail order rate. Eventually, in a diminishing cycle, the factory settles into a new equilibrium of supply and demand. This automatic re-alignment of production is a good example of error correcting by a balancing feedback loop. Factory managers do not need to know the exact real value of worker productivity in order to hire the right number of workers. If they overestimate productivity then finished inventory will be below desired and the resulting correction for inventory will create pressure to hire more workers, thereby compensating for the mis-estimation of productivity.
Desired workforce brings us to the end of the workforce sector formulations. In the rest of the chapter, we continue our simulation analysis with scenarios and what-ifs typical of the kind of work conducted in phase 2 of a modelling project.
Figure 5.23. Equations for workforce planning
