**A College Enrollment Model: Part II**

To study the effects of admissions and transfer policies, generalize the enrollment model in Example 4.8-1 to allow for varying admissions and transfers .

•** Solution**

Leta(k) be the number of new.freshmen admitted in the spring of year k for the following year k + 1 and let d(k) be the number of transfers into the following year’s sophomore class. Then the model becomes

where we have written the coefficients C2l, C22, and so on in symbolic, rather than numerical, form so that we can change their values if desired. This model can be represented graphically by a state transiiion diagram, like the one shown in Figure 4.8-l. Such diagrams are widely used to represent time-dependent and probabilistic processes. The arrows indicate how the model’s calculations are updated for each new year. The enrollment ar year k is described completely by the values of xI(k),

x2(k), x3{k), and x4(k); that is, by the’vector x(k), which is called the state, vector. The elements of the state vector are the state variables. The state transition diagram shows how the new values of the state variables depend on both the previous values and the inputs a(k) and d(k).

Suppose that the initial total enrollment of 1480 consists of 500 freshmen,

AOO sophomores, 300 juniors, and 280 seniors. The college wants to study, over a 10- year period, the effects of increasing admissions by 100 each year and transfers by 50 each year until the total enrollment reaches 4000; then admissions and transfers will be, held constant. Thus the admissions and transfers for the next 10 years are given by

a(k) = 900 + lOOk

d(k) = 150 + 50k

for k = 1,2,3, … until the college’s total enrollment reaches 4000; then admissions and transfers are held constant at the previous year’s levels. We cannot determine when this event will occur without doing a simulation. Table 4.8-1 gives the pseudocode for solving this problem. The enrollment matrix E is a 4 x 10 matrix whose columns represent the enrollment in each year.

Because we know the length of the study (10 years), a for loop is a natural choice. We use an if statement to determine when to switch from the increasing admissions and transfer schedule to the constant schedule. A MATLAB script file to predict the enrollment for the next 10 years appears in Table 4.8-2. Figure 4.8-2 shows

the resulting plot. Note that after year 4 there are more sophomores than freshmen; The reason is that the increasing transfer rate eventually overcomes the effect of the increasing admission rate. In actual practice this program would be run many times to analyze the effects of different admissions and transfer policies and to examine what happens if different values are used for the coefficients in the matrix C (indicating different dropout and repeat rates).

**Test Yo”r Understanding**

14.8-2 IIi the program in Table 4.8-2, lines 16 and 17 compute the values of a (k) and d (k) . These lines are repeated here:a(k) = 900+100*k

d(k) = 150+50*ki

Why does the program contain the line a ( 1) = 1000; d (1) = 200 ; ?