5.3. Equation Formulations and Computations in Production Control
It is common for equation formulations to be organised in sectors and blocks that correspond as closely as possible to the visual model. We begin with inventory accumulation as shown in Figure 5.10. These are the standard equations for a stock and flow network. Refrigerator inventory at time t is equal to refrigerator inventory at the previous point in time t−1, plus the net change resulting from production (the inflow) and shipments (the outflow) over the interval dt between t−1 and t. The initial inventory is set at 4 000 refrigerators, deliberately chosen so that there are four weeks coverage of initial shipments, exactly in line with the factory's rule of thumb. The production rate is identically equal to desired production, a temporary but convenient assumption that says the factory can always produce as many refrigerators as needed at just the right time. The shipment rate is equal to the retail order rate implying there is enough finished inventory available in the shipping bay to fill orders.[] The retail order rate itself is formulated as a one time step increase, a simple yet insightful exogenous change. The order rate begins at 1 000 refrigerators per week and then, in week 10, rises by 10 per cent (100 refrigerators per week).
[] In practice some factories temporarily run out of inventory and/or deliberately choose to build to order. Such supply bottlenecks are usually modelled by introducing an order backlog in parallel with finished inventory. Retail orders then flow into the backlog. Both inventory and backlog are simultaneously reduced by the shipment rate, which is itself jointly a function of the size of the backlog and finished inventory. This is a classical dynamical representation of supply and demand. A typical formulation for the shipment rate is:
shipment rate = desired shipment rate * effect of inventory on shipments
desired shipment rate = backlog/normal delivery delay
effect of inventory on shipments = GRAPH (adequacy of finished inventory)
adequacy of finished inventory = finished inventory/desired finished inventory
Here the effect of inventory on shipments is a non-linear function that depends on the adequacy of finished inventory. The typical shape of the curve is convex – steep to begin with and then gradually flattening out. If there is no inventory at all then there are no shipments. With a small amount of well-chosen inventory, the factory can fill orders for its high volume products. With plenty of inventory the factory can ship everything it wants to. It is a useful exercise to extend the production and workforce model by adding a backlog and by modifying the shipments formulation along the lines outlined above.
Figure 5.10. Equations for inventory accumulation
5.3.1. Forecasting Shipments – Standard Formulations for Information Smoothing
The formulations for forecasting shipments are shown in Figure 5.11. They are a specific example of standard formulations for 'information smoothing' that occur whenever people in organisations perceive, estimate or formally measure the conditions around them. In this case, factory managers are estimating future demand. We already know the factory regards past shipments as a reliable guide to future demand, but how is this idea expressed algebraically? At the heart of the formulations is a stock accumulation called the 'average ship rate' and a procedure for updating it. The average ship rate at time t (the current estimate of demand) depends on the average ship rate at time t−1, modified or updated by the 'change in average ship rate' over the interval dt. The size of the update depends on the gap between the shipment rate and the average ship rate, divided by the 'time to average ship rate'. What does this really mean? The gap signifies change. The bigger the gap, the more demand has changed relative to its past average, and so the greater the scope for updating the forecast. But the speed of update is also important. A large gap may just be a temporary blip and so factory managers may not want to fully adjust their estimate of demand. This wait-and-see idea is captured by the 'time to average ship rate'. If the averaging time is short then the update is swift, and if the averaging time is long the update is gradual. In this particular case, the averaging time is set at eight weeks. Hence, computationally the average ship rate is updated by one-eighth per week of whatever gap exists between the shipment rate and the average ship rate.
Figure 5.11. Forecasting shipments through information smoothing
Conceptually this means the factory forecast ignores short-term blips in demand but takes full account of systematic changes that persist over several months. The formulation is subtle and remarkably general. It can represent the natural smoothing that takes place in any process of monitoring or measurement (for example, in a thermometer used to measure air temperature where there is a noticeable delay in the reading if the thermometer is moved outside on a frosty day). Equally, the formulation can represent psychological smoothing that people apply to unfolding events (Makridakis, Chatfield, Hibon et al., 1993). They tend to give most weight to recent events still vivid in memory and less to those in the distant past. Technically, the formulation corresponds exactly to so-called 'exponential averaging', where the average is computed from the sum of past observations, giving most weight to recent observations and progressively less to older ones, with a weighting pattern that decays exponentially (for a proof of this weighting pattern, see Forrester 1961, appendix E on smoothing of information and Sterman 2000, Chapter 11, section 11.3 on information delays).
5.3.2. Inventory Control – Standard Formulations for Asset Stock Adjustment
The equations for inventory control are shown in Figure 5.12. Here is another standard and classic collection of formulations. They can be used when modelling any kind of purposive, goal-directed behaviour in organisations. At the heart of these formulations is a vital distinction between the actual condition of an organisational asset and the desired condition. These two conditions can exist side-by-side and need not be the same. In practice, they differ most of the time and whenever they differ there is pressure for corrective action. In this case, the control or management of refrigerator inventory is of central interest. Factory managers know how much inventory they would like to be holding (the desired inventory) and can measure how much is currently in the factory (refrigerator inventory). A gap calls for corrective action, either cutting production to eliminate a surplus or increasing production to remedy a shortfall. The correction for inventory depends on the difference between desired inventory and refrigerator inventory. That much is intuitively obvious, but the inventory gap alone does not say how much production should change in response. If the factory finds itself 2 000 refrigerators short, it could (in principle) schedule all 2 000 units for production in one week or it could build 200 per week for 10 weeks, or it could even ignore the gap entirely. This managerial sense of urgency is captured in the parameter 'time to correct inventory', which is set at a moderate value of eight weeks. The correction for inventory is equal to the inventory gap divided by the time to correct inventory. Desired inventory is formulated as the product of the average ship rate and normal inventory coverage. The coverage is set at four weeks consistent with the factory's rule of thumb to avoid 'stockouts'. Notice that the dimensions of all the equations balance properly.
Figure 5.12. Equations for inventory control
5.3.3. Desired Production
Desired production is formulated as the sum of the average ship rate and correction for inventory. The two main pressures on production combine as shown in Figure 5.13. Not surprisingly, factory managers anchor their production plan to the average ship rate, which is their estimate or forecast of demand. They also adjust the plan to take account of any surplus or shortage of refrigerators signalled by inventory control.
Figure 5.13. Desired production
5.3.4. The Computations Behind Simulation
So far, we have described the formulations for production control. Before moving on to consider the workforce, I will use the existing small algebraic model to demonstrate the computations that take place behind the scenes of any simulator. The model is the same one that generated simulations of the ideal flexible factory earlier in the chapter, which, as we saw, easily copes with an unexpected boom in orders. Nevertheless, there are some interesting dynamics associated with the depletion and rebuilding of inventory and the amplification of production. It is the computation of these dynamics I will now demonstrate.
At the start of any simulation, the initial values of all the stock variables are known. Collectively they determine the state of the system in much the same way that the levels of fuel, oil, coolant, screen wash and brake fluid tell you something about the state of your car. These stocks are connected to their inflows and outflows through the model's feedback loops. So, knowing the initial values, it is possible to work out the changes that will inevitably occur in the stocks during the coming interval of time. The stocks are updated according to their prescribed inflows and outflows to yield a new state of the system at the next point in time. By stepping through time (finely sliced) and repeating these calculations over and over again, the trajectories of the model's variables are made apparent.
To illustrate the procedure, consider refrigerator inventory in the production control model as shown in Figure 5.14. The equation at the top of the diagram is the standard equation for stock accumulation that shows how refrigerator inventory is updated from one point in time (t−1) to the next (t) according to the inflows and outflows over the interval dt between t−1 and t. Notice there are 4000 refrigerators at time zero to prime the calculation engine. In a larger model, it is necessary to provide initial values for every single stock. Time is sliced into tiny intervals of dt. This demarcation is a purely technical and computational artefact that overlays the time units in which the model is developed. In the production model, factory time is in weeks and dt is set to half a week, small enough to ensure numerical accuracy of the simulation.[] The calculation proceeds in three stages. The stock is assigned its initial value of 4 000 at the start of the simulation and then updated according to the inflow of production and the outflow of shipments over the coming interval or time step dt. Rates of change for the coming time step depend on the value of stocks and converters (also known as auxiliaries) at the end of the current period. In this case, the production rate is equal to the desired production rate and the shipment rate is equal to the retail order rate. Converters (auxiliaries) themselves depend on stocks and other auxiliaries according to the relationships described in the information feedback network. (Note that the meaning of the term 'auxiliary' is computational – it refers to variables that are auxiliary to the calculation of flow rates.) Such mutual dependence is a natural consequence of feedback. The one exception is when a variable is assumed to be exogenous (not in a feedback loop) – such as the retail order rate.
[] As a rule of thumb, dt should be no more than one quarter of the shortest time constant in the model. In the production control model, dt is half a week and there are two time constants (time to average ship rate and time to correct inventory) both of eight weeks. So dt is one-sixteenth of the smallest time constant, comfortably within the rule.
Figure 5.14. The computation process and time slicing
Now we re-simulate the production control model to see the calculations step by step. Figure 5.15 contains a table of numerical values for refrigerator inventory, production rate and shipment rate over the period zero to 15 weeks. There is also a small time chart that shows the resulting trajectories all the way to week 80 (note this is only half the duration of the original simulation, so the trajectories appear stretched). Refrigerator inventory starts at 4 000 and remains in equilibrium with production rate exactly equal to the shipment rate of 1 000 refrigerators per week. In week 10, shipment rate rises to 1100 mimicking the exogenous one-time increase in the retail order rate. The table shows the numerical consequences of this uplift in demand. The production rate builds slowly from a value of 1 000, reaching 1 104 by week 15. These particular week-by-week numbers are dictated by desired production whose computation is shown in the next figure. For five weeks, the refrigerator inventory falls because shipments exceed production.
Meanwhile, the computations within the information network are proceeding as shown in Figure 5.16. Here the focus is on weeks 10–15. During this five week interval, the shipment rate (2nd column) remains steady at 1 100 refrigerators per week, in line with demand. The remaining columns show the numbers behind desired production. The average ship rate (3rd column) increases from 1 000 to 1 048.71 refrigerators per week, which is simply the numerical consequence of the assumed exponential smoothing. (Incidentally, simulators often display numbers with an unnecessary degree of precision and it is up to the modeller to interpret them appropriately. In this case, we are seeing the result of judgemental forecasting and it is best to round to the nearest whole number of refrigerators. The same common sense applies to the interpretation of numbers in the other columns and so I will report whole numbers.) Desired inventory (4th column) is defined as four times the average ship rate and the numbers reflect this rule of thumb, rising from 4000 to 4195 as factory managers re-appraise demand. The correction for inventory (5th column) is defined as one-eighth of the difference between the desired inventory and the refrigerator inventory. This fragment of algebra is repeated above the table using the abbreviation dit for desired inventory at time t and rit for refrigerator inventory at time t. In week 10, desired inventory is 4 000, exactly the same as the refrigerator inventory in week 10 from Figure 5.15. So the correction for inventory is zero. By week 11, desired inventory has risen to 4 050 refrigerators while the actual number of refrigerators has fallen to 3 900, a gap of 150. This gap calls for an additional 19 (150/8) refrigerators per week. Hence, at the start of week 11, desired production (6th column) is 1 031 refrigerators per week: 1 012 from the average ship rate (asrt) and 19 from the correction for inventory (cit). It is this computed volume, still far below the shipment rate of 1 100, that drives the production rate in week 11. As a result, refrigerator inventory (in Figure 5.15) falls to 3 831 by the start of week 12.
Figure 5.15. The mechanics of simulation and stock accumulation
Figure 5.16. Computations in the information network
The next round of calculations then takes place. In week 12, the shipment rate is 1 100 and the average ship rate is 1 023. Desired inventory is now 4 094 (slightly more than 4*1 023 due to rounding) and correction for inventory is 33 refrigerators per week (one-eighth of the difference between 4 094 and 3 831). Desired production is therefore 1 056 refrigerators per week (1 023 + 33) and this becomes the volume of production throughout week 13. Production is higher than the previous week, but still lower than the shipment rate of 1 100, so inventory falls to 3 788. The calculations continue into week 13 with an average ship rate of 1 033, desired inventory of 4 132, correction for inventory of 43, and desired production of 1076, and so on. By continuing the computations to year 80, the two time charts in Figure 5.16 are created. So much for computation, now we develop the rest of the model.
Figure 5.17. Stock and flow diagram for workforce management
