System Feedback Loops

Time Warp 3 and CVP Analysis

How many feedback loops does a given system have? Unfortunately, there are no general formulas for finding the number of loops for a given system, short of direct enumeration. We can, however, specify an upper bound on the number of loops by considering the maximally connected system, i.e. one in which every flow x? depends upon every state variable x and every auxiliary variable y, and every y depends upon every x and 'as many as possible' of the other auxiliary variables y. We show in the appendix (Lemma A.1) that the number of loops Sn,p in a maximally connected system with n state variables and p auxiliary variables grows as 2np(n - 1)!. One might ask whether realistic models, which clearly have far fewer links, may yet contain a manageable number of loops.

While the number is certainly much smaller than Sn,p, it is still substantial—large enough to pose problems for the practitioner. For instance, the World3 model has 5 state variables, 44 auxiliaries and 76 links between variables. The model contains 81 non-reduced loops, and 41 reduced loops. Nathan Forrester's macroeconomic model) (in it's basic version) has 8 endogenous state variables, 18 auxiliary variables, and 44 links.

The model contains 39 non-reduced and 118 reduced loops. Clearly, the size of the independent loop set is equal to the rank of the full cycle matrix. Moreover, the set is not unique. In a moment, we shall consider the size of the independent loop set and how to construct such a set. Before proceeding, however, we consider the signs of the gains as well. Even though an independent loop set can be assigned arbitrary absolute gains and changes in gains by an appropriate assignment or change in edge gains, the signs of the loop gains cannot always be determined independently.

The independence of signs is determined by the binary rank ?2 of Cr. In the appendix, we show that ?2 = ? and provide an example of an independent loop set where ?2 < ?. However, the procedure in Theorem 2.1 to select an independent loop set, also assures that the gains of the loops in the set can be assigned arbitrary signs.

CVP analysis can be used in several different ways; to calculate the breakeven point, the price required to earn particular profits or the units of product that need to be sold to earn that profit. Since our main concern in this task to perform better than Joe Schmoe, we will focus on developing a strategy which results in more profits for the company than under Joe Schmoe which was $ 954,830,241 from years 2012 to 2016. So our focus will be to develop a strategy which results in a total profit of $ 1,000,000,000. We achieve this profit over the period of four years as shown in the table below. This allocation has been made after analyzing the profits under Joe Schmoe and Time warp 1 and hence the figures ...