BSc (Hons) Software Engineering BSc (Hons) Computer Science with Network Security Cohorts BCNS/ 06 / Full Time & BSE/ 06 / Full Time Resit Examinations for 2008-2009 / Semester 1 Examinations for 2008-2009 / Semester 2 MODULE: Operations Research MODULE CODE: MATH 3113 Duration: 2 Hours 30 Minutes Instructions to Candidates: 1. Answer ALL the questions. 2. Questions may be answered in any order but your answers must show the question number clearly. 3. Always start a new question on a fresh page. 4. All questions carry equal marks. 5. Total marks 100. This question paper contains 4 questions and 6 pages. Page 1 of 6
ANSWER ALL THE QUESTIONS Question 1: (25 Marks) Unsatisfactory control of spare parts in a particular mechanical workshop is resulting in high carrying costs for some items and high stock-out costs for others. A study of the past demand and order lead time patterns for a particular item, X, gives the following information: Daily demand 5 6 7 8 9 10 11 12 13 Probability 0.20 0.19 0.15 0.10 0.09 0.08 0.07 0.06 0.06 Probability of a lead time of 2 days is 0.10 Probability of a lead time of 3 days is 0.26 Probability of a lead time of 4 days is 0.40 Probability of a lead time of 5 days is 0.24 A lead time of x days means the order is placed at the end of the day that stock runs out and is received after x full days. A proposal has been made that a policy of ordering a basic 35 units should be adopted whenever stock falls below 30 units. To the order of 35 units there should be added the number of units necessary to bring stock up to the re-order point of 30 units at the time the order is made. A stock of 40 units is in hand. Ordering costs amount to $25 per order, carrying costs are $1 per day and stock-out costs are $4 per unit. Stock-out costs are incurred automatically whenever there is no stock in store, since the stock has to be procured immediately from an outside source. The workshop administrators decided to carry out a Monte Carlo simulation of the demand for the item X. Random numbers for 10 days of the demand sequence are given below: Page 2 of 6
Demand 85 80 54 90 25 20 41 15 73 10 Lead time 15 01 Carry out the simulation of the 10 days demand on stock levels and calculate the average daily demand from the simulation. (19 Marks) Compare the original expected daily demand with the average daily demand obtained from the simulation. (4 Marks) (c) Calculate the average daily stock cost. Question 2: (25 Marks) The following payoff table shows profit for a decision analysis problem with two decision alternatives and three states of nature. State of Nature Decision Alternative S 1 S 2 S 3 D 1 250 100 25 D 2 100 100 75 Construct a decision tree for this problem. (5 Marks) If the decision maker knows nothing about the probabilities of the three states of nature, what is the recommended decision using the optimistic, pessimistic and minimax regret approaches? (6 Marks) Page 3 of 6
(c) Suppose that the decision maker has obtained the probability assessments: P (S 1 ) = 0.65, P (S 2 ) = 0.15, and P (S 3 ) = 0.20. (1) Use the expected value approach to determine the optimal decision. (5 Marks) (2) What is the optimal decision strategy if perfect information were available? (3 Marks) (3) What is the expected value for the decision strategy developed in part (2)? (4) Using the expected value approach, what is the recommended decision without perfect information? What is the expected value? (5) What is the expected value of perfect information? Question 3: (25 Marks) Consider stochastic matrix A = ( 0 1 0.4 0.6 ). (1) Is the matrix A ergodic? Justify your answer. (2) Compute lim n An. (3 Marks) (5 Marks) A camera store stocks a particular model camera that can be ordered weekly. Let D 1, D 2, represent the demand for this camera (the number of units that would be sold if the inventory is not depleted) during the first week, second week,..., respectively. It is assumed that the D i are independent and identically Page 4 of 6
distributed random variables having a Poisson distribution with a mean of 1. Let X 0 represent the number of cameras on hand at the outset, X 1 the number of cameras on hand at the end of week 1, X 2 the number of cameras on hand at the end of week 2, and so on. Assume that X 0 = 3. On Saturday night the store places an order that is delivered in time for the next opening of the store on Monday. The store uses the following order policy: If there are no cameras in stock, the store orders 3 cameras. However, if there are any cameras in stock, no order is placed. Sales are lost when demand exceeds the inventory on hand. (1) Show that the stochastic process {X t } (t = 0, 1, 2,...) is a Markov chain. (2) Construct the one-step transition matrix, P. (3) Construct the state transition diagram. (3 Marks) (11 Marks) (4) Given that P 6 = 0.2856 0.2846 0.2632 0.1666 0.2863 0.2849 0.2628 0.1661 0.2857 0.2848 0.2633 0.1662 0.2856 0.2846 0.2632 0.1666. State the probability that there will be no cameras in stock 6 weeks later given that there is one camera left in stock at the end of a week. (1 Marks) Question 4: (25 Marks) A person has 3 units of money available for investment in a business opportunity that matures in 1 year. The opportunity is risky in that the return is either double or nothing. Based on past performance, the likelihood of doubling one s money is 0.6, while the chance of losing an investment is 0.4. Money earned one year can be reinvested in a later year and investments are restricted to unit amounts. Page 5 of 6
When dynamic programming is used to find the investment strategy for the next 4 years that will maximize the expected total holdings at the end of that period, the problem is formulated as a four-stage process with each stage representing a year. The states s j are the amounts of money available for investment for stage j (j = 1, 2, 3, 4). Let f j (s j ) denote the maximum expected holdings at the end of the process, starting in state s j at stage j. By clearly explaining your reasoning show that a recursive formula for finding the maximum expected holdings at the end of four years is given by [ ] f j (s j ) = max αf j+1 (s j + x j ) + βf j+1 (s j x j ), x j =0, 1,, s j for j = 1, 2, 3 and 4, where the values of α and β are to be determined. (12 Marks) Write down an expression for f 5 (s). (1 Marks) (c) Find the maximum expected holdings at the end of the four years. (12 Marks) ***END OF PAPER*** Page 6 of 6