Solved Problems
Solved Problem 14-1 George Walls, president of Bradley School, is concerned about declining enrollments. Bradley School is a technical college that specializes in training computer programmers and computer operators. Over the years, there has been a lot of competition among Bradley School, International Technology, and Career Academy. The three schools compete in providing education in the areas of programming, computer operations, and basic secretarial skills.
To gain a better understanding of which of these schools is emerging as a leader, George decided to conduct a survey. His survey looked at the number of students who transferred from one school to the other during their academic careers. On average, Bradley School was able to retain 65% of those students it originally enrolled. Twenty percent of the students originally enrolled transferred to Career Academy, and 15% transferred to International Technology. Career Academy had the highest retention rate: 90% of its students remained at Career Academy for their full academic program. George estimated that about half the students who left Career Academy went to Bradley School, and the other half went to International Technology. International Technology was able to retain 80% of its students after they enrolled. Ten percent of the originally enrolled students transferred to Career Academy, and the other 10% enrolled in Bradley School.
Currently, Bradley School has 40% of the market. Career Academy, a much newer school, has 35% of the market. The remaining market share—25%—consists of students attending International Technology. George would like to determine the market share for Bradley for the next year. What are the equilibrium market shares for Bradley School, International Technology, and Career Academy?
Solution
The data for this problem are summarized as follows:
State 1 initial share 0.40—Bradley School
State 2 initial share 0.35—Career Academy
State 3 initial share 0.25—International Technology
The transition matrix values are
TO 1 2 3 FROM BRADLEY CAREER INTERNATIONAL 1 BRADLEY 0.65 0.20 0.15 2 CAREER 0.05 0.90 0.05 3 INTERNATIONAL 0.10 0.10 0.80 For George to determine market share for Bradley School for next year, he has to multiply the current market shares times the matrix of transition probabilities. Here is the overall structure of these calculations:
Thus, the market shares for Bradley School, International Technology, and Career Academy can be computed by multiplying the current market shares times the matrix of transition probabilities, as shown. The result will be a new matrix with three numbers, each representing the market share for one of the schools. The detailed matrix computations follow:
Now George would like to compute the equilibrium market shares for the three schools. At equilibrium conditions, the future market share is equal to the existing or current market share times the matrix of transition probabilities. By letting the variable X represent various market shares for these three schools, it is possible to develop a general relationship that will allow us to compute equilibrium market shares.
Let
At equilibrium,
The next step is to make the appropriate multiplications on the right-hand side of the equation. Doing this will allow us to obtain three equations with the three unknown X values. In addition, we know that the sum of the market shares for any particular period must equal 1. Thus, we are able to generate four equations, which are now summarized:
Because we have four equations and only three unknowns, we are able to delete one of the top three equations, which will give us three equations and three unknowns. These equations can then be solved using standard algebraic procedures to obtain the equilibrium market share values for Bradley School, International Technology, and Career Academy. The results of these calculations are shown in the following table:
SCHOOL MARKET SHARE (Bradley) 0.158 (Career) 0.579 (International) 0.263 Solved Problem 14-2 Central State University administers computer competency examinations every year. These exams allow students to “test out” of the introductory computer class held at the university. Results of the exams can be placed in one of the following four states:
State 1: pass all of the computer exams and be exempt from the course
State 2: do not pass all of the computer exams on the third attempt and be required to take the course
State 3: fail the computer exams on the first attempt
State 4: fail the computer exams on the second attempt
The course coordinator for the exams has noticed the following matrix of transition probabilities:
Currently, there are 200 students who did not pass all of the exams on the first attempt. In addition, there are 50 students who did not pass on the second attempt. In the long run, how many students will be exempted from the course by passing the exams? How many of the 250 students will be required to take the computer course?
Solution
The transition matrix values are summarized as follows:
TO FROM 1 2 3 4 1 1 0 0 0 2 0 1 0 0 3 0.8 0 0.1 0.1 4 0.2 0.2 0.4 0.2 The first step in determining how many students will be required to take the course and how many will be exempt from it is to partition the transition matrix into four matrices. These are the I, 0, A, and B matrices:
The next step is to compute the fundamental matrix, which is represented by the letter F. This matrix is determined by subtracting the B matrix from the I matrix and taking the inverse of the result:
We first find
Now multiply the F matrix by the A matrix. This step is needed to determine how many students will be exempt from the course and how many will be required to take it. Multiplying the F matrix times the A matrix is fairly straightforward:
The final step is to multiply the results from the FA matrix by the M matrix, as shown here:
As you can see, the MFA matrix consists of two numbers. The number of students who will be exempt from the course is 231. The number of students who will eventually have to take the course is 19.