Markov Analysis

Graphic Era University,Dehradun University,Dehradun

Assignment Markov Analysis Ques-1 A grocer stocks his store with three types of detergent A, B and C. when brand A is sold out the probability is 0.7 that he stocks up with brand A again. When he sells out brand B the probability is 0.8 that he will stock up again with brand B. when he sells out brand C the probability is 0.6 that he will stock up with brand C again. When he switches to another detergent he does so with equal probability for the remaining two brands. Find the transition matrix. In the long run, how does he stock up with detergent? Ans A=0.3077, B=0.4615 and C=0.2308. Ques-2 On January 1 (this year) ,Bakery A has 40% of its local market share while Bakery B and C had 40 and 20%, res., of the market share. Based on market research Bakery A retains 90% and gain 5%of B’s customer and 10% of C’s customer. Bakery B retain 85% and gain 5% of A’s customer and 7% of C’s customer. Bakery C retain 83% and gain 5% of A’s customer and 10% of C’s customer. What will each firm’s share be on January 1 next year and what will each firm’s market share be at equilibrium? (Ans Bakery A=43%, Bakery B=28%, Bakery C=29%) Ques-3 A grocer stocks his store with 3 types of detergents A,B and C. when brand is sold out the probability is 0.7 that he stocks up with brand A again. When he sells out brand B the probability is 0.8 that he will stock up again brand. Finally, when he sells out brand C the probability is 0.6 that he will stock up with brand C again. When he switches to another detergent he does so with equal probability for the remaining 2 brands. Find the transition matrix. In the long run, how does he stock up with detergents? Answer-a) 0.3077, b) 0.4615, c) 0.2308

Ques-4 A manufacturing company has a certain piece of equipment that is inspected at the end of each day and classified as just overhauled, good, fair or inoperative. If the item is inoperative it is overhauled, a procedure that takes one day. Let us denote the four classifications as states 1, 2, 3, and respectively. Assume that the working condition of the equipment follows a Markov chain with the following transition matrix:

Tommorrow

P = Today

Prepared By:-Sanjeev Arora (Faculty Management)

1

2

3

4

1

0

¾

¼

0

2

0

½

½

0

3

0

0

½

½

4

1

0

0

0

Graphic Era University,Dehradun

If it costs Rs. 125 to overhaul a machine (including lost time) on the average and Rs. 75 as Production lost if a machine is found inoperative. Using the steady-state probabilities, compute the expecte per day cost of maintenance. (Ans = Rs.36.36) Ques-5 The ‘School of international studies for Population ‘ found out by its survey that the mobility of the population [in per cent] of a state to a village, town, and city is in the following percentages. To Village Town City Village 50 30 20 Town 10 70 20 City 10 40 50

What will be the proportion of population in village, town, city after two years, given that the present population has proportions of 0.7,0.2, 0.1 in the village, town and city respectively? Ans:[0.252, 0.479, 0.269]

Ques-6 A professor tries not to be late for class too often. If he is late one day, he is 90%sure to be on time next time. If he is on time then the next time there is 30% chance of his being late. In the long run, how often is he late for class.Ans.0.25, 0.75 Ques-7 Ray Cahnman is the proud owner of a 1955 sports car.On any given day, Ray never knows whether his car will start. Ninety percent of the time it will start if it started the previous morning, and 70% of the time it will not start if it did not start the previous morning. (a) Construct the matrix of transition probabilities. (b) What is the probability that it will start tomorrow if it started today?(90%) (c) What is the probability that it will start tomorrow if it did not start today?(30%)

Ques-8 Over any given month, Dress-Rite loses 10% of its customers to Fashion, Inc., and 20% of its market to Luxury Living. But Fashion, Inc., loses 5% of its market to Dress-Rite and 10% of its market to Luxury Living each month; and Luxury Living loses 5% of its market to Fashion, Inc., and 5% of its market to Dress-Rite. At the present time, each of these clothing stores has an equal share of the market. What do you think the market shares will be next month? (Ans. 4/15, 5/15, 6/15) What will they be in three months?(19.52%,32.52%,47.96%) Draw tree diagrams to illustrate next month Ques-9 Professor Green gives two-month computer programming courses during the summer term. Students must pass a number of exams to pass the course, and each student is given three chances to take the exams. The following states describe the possible situations that could occur: 1. State 1: pass all of the exams and pass the course Prepared By:-Sanjeev Arora (Faculty Management)


2. State 2: do not pass all of the exams by the third attempt and flunk the course 3. State 3: fail an exam in the first attempt 4. State 4: fail an exam in the second attempt After observing several classes, Professor Green was able to obtain the following matrix of transition probabilities:

At the present time, there are 50 students who did not pass all exams on the first attempt, and there are 30 students who did not pass all remaining exams on the second attempt. How many students in these two groups will pass the course, and how many will fail the course? (61, 19) Ques-10 An accounts receivable system normally places debts or receivables from its customers into one of several categories or states depending on how overdue the oldest unpaid bill is. Of course, the exact categories or states depend on the policy set by each company. Four typical states or categories for an accounts receivable application follow: State 1 : paid, all bills State 2 : bad debt, overdue more than three months State 3 : overdue less than one month State 4 : overdue between one and three months

Assume that there is Rs.2,000 in the less than one month category and Rs.5,000 in the one to three month category. How much amount will eventually be paid up and how much will be become bad debts ( Rs. 6240,Rs.760) Ques-11 The University of South Wisconsin has had steady enrollments over the past five years. The school has its own bookstore, called University Book Store, but there are also three private bookstores in town: Bill’s Book Store, College Book Store, and Battle’s Book Store. The university is concerned about the large number of students who are switching to one of the private stores. As a result, South Wisconsin’s president, Andy



Lange, has decided to give a student three hours of university credit to look into the problem. The following matrix of transition probabilities was obtained:

At the present time, each of the four bookstores has an equal share of the market. What will the market shares be for the next period? If the same conditions exist, what long-run market shares can Andy expect for the four bookstores? Ans:- (18.75, 26.25, 0.30, 0.25 , 12.90%,25.8%,41.94%,19.35%)


Markov Analysis

Recommend Documents