7.11. Appendix – Gain of a Reinforcing Loop

Gain is a technical term used to describe the strength of a reinforcing loop. The procedure for calculating gain is outlined in Figure 7.34. There is a reinforcing loop A-B-C-D. What it represents is immaterial. What matters is whether the causal links amplify or attenuate a change that originates in the loop. To test this property we first imagine the loop is cut at some convenient point and then unfolded to form a linear chain or 'open loop'. In this case a cut is made in the link connecting D and A. The gain of the open loop is defined as the size of the change transmitted back to A when the value of A itself changes by one unit. Alternatively, if A changes by a small amount ΔA, then the gain is the size of the change transmitted back to A divided by ΔA. The calculation assumes a steady state is achieved, so enough time elapses for any stock adjustment processes to run their course. The full name for the calculation is the 'open loop steady state gain'.

Figure 7.34. The gain of a reinforcing loop

The concept is best illustrated with a practical example. Figure 7.35 shows the sales growth loop from the market growth model, including the parameters that determine the strength of causal links. In this case, imagine the loop is cut at a point in the link between revenue and sales budget. Suppose there is a small increase in the sales budget Δsb. The size of the change transmitted back to the sales budget is defined in the model as the change in the order fill rate (Δofr) multiplied by the product price (pp) and the fraction of budget to marketing (fbm). By working back through the links we can express the change in the order fill rate Aofr itself as a function of the change in the sales budget Asb and so end up with an expression for gain that depends only on the parameters around the edge of the loop. The steps in the calculation are shown below:

Figure 7.35. Parameters affecting the gain of the sales growth loop

Step 1: gain = Δofr * pp * fbm/Δsb

Step 2: gain = Δco * pp * fbm/Δsb, since in steady state the change in the order fill rate Δofr is equal to the change in customer orders Aco.

Step 3: gain = Δsf * sfp * edd * pp * fbm/Δsb, because the change in customer orders Δco is equal to the change in sales force Δsf multiplied by sales force productivity.

Step 4: gain = Δsb * sfp * edd * pp * fbm/ss * Δsb, since in steady state the change in sales force Δsf is equal to the change in the sales budget Δsb divided by sales force salary ss.

Step 5: gain = sfp * edd * pp * fbm/ss, cancelling the original change in sales budget Δsb to leave only numerical parameters.

There are five parameters that determine the open loop steady state gain: sales force productivity (sfp), the effect of delivery delay (edd), product price (pp), the fraction of budget to marketing (fbm) and sales force salary (ss). Substituting numerical values from the model gives a gain of 2.4 (10* 1* 9600 * 0.1/4000). Since the gain is greater than one the loop generates growth. Notice the effect of delivery delay (edd). We assume in the calculation that it takes a neutral value of one. However in the full model the effect is variable and can take a value anywhere between one and zero, depending on delivery delay recognised by the customer (as shown in Figure 7.16). If the effect of delivery delay (edd) were to take a value of 0.4167 (which is the inverse of 2.4), then the gain would be exactly one and customer orders would remain steady and neither increase nor decline. If the effect of delivery delay were to take a value of say 0.2 then the gain would be less than one and customer orders would go into a spiral of decline. This transformation from growth to decline is exactly what happens in the base case simulation of the full model as shown in Figure 7.27.