PROGRAMME MBA-Human Resources & knowledge Management MBA- Project Management Master of Business Administration General MBA-Marketing Management COHORT MBAHR/11/PT MBAPM/11/PT MBAG/11/PT MBAMM/11/PT Examinations for 2010 2011 Semester II / 2011 Semester I MODULE: OPERATIONS RESEARCH MODULE CODE: MBA 1902 DURATION: 3 HOURS Instructions to Candidates: 1. This question paper consists of Section A and Section B. 2. Section A is compulsory. 3. Answer any three questions from Section B. 4. Always start a new question on a fresh page. 5. Total Marks: 100. This question paper contains 6 questions and 10 pages. Page 1 of 10
SECTION A: COMPULSORY QUESTION 1: (40 MARKS) Part A An oil refinery produces three different types of gasolene, A, B and C. Two important characteristics of gasolene are its performance number (PN) and its vapour pressure (RVP). The table below gives these characteristics and the production cost per barrel for each type of gasolene. Gasolene PN RVP Production cost per barrel ------------------------------------------------------------------- A 98 5 $4.50 B 103 8 $5.20 C 108 6 $5.50 The company wishes to blend these three types of gasolene to produce an aviation fuel. Such fuels must have a PN of at least 100 and a RVP of at most 7. The PN and RVP of a gasolene blend is the weighted average of its constituents. (a) Show that the problem of determining the proportions in which the three types of gasolene A, B and C should be blended to produce an aviation fuel satisfying the PN and RVP requirements at minimum cost can be modeled by the following linear programming problem P: Find x 1, x 2, x 3 є R to minimize subject to z = 4.5x 1 + 5.2x 2 + 5.5x 3 x 1 + x 2 + x 3 = 1 98x 1 + 103x 2 + 108x 3 100 5x 1 + 8x 2 + 6x 3 7 Take care in your answer to define the decision variables and to explain briefly how the objective function and the constraints are derived. (5 marks) Page 2 of 10
(b) Use the equality constraint to eliminate the variable x 3 from P. State carefully the revised problem with two decision variables. (c) On the graph paper provided, plot the constraint lines for the revised problem with two decision variables. Hence, or otherwise, identify the feasible corner point at which z achieves its minimum value and determine the optimum values of x 1 and x 2. (8 marks) (d) Interpret this solution as a schedule for the oil refinery, giving the production cost per barrel, the PN and the RVP of this blend of aviation fuel. Part B (a) A shopkeeper buys newspapers from the delivery truck at the beginning of the day. During the day, he sells newspapers. Leftover newspapers at the end of the day are worthless. Assuming that each newspaper costs Rs 10.00 and sells for Rs 18.00 and that the following probability distribution is known. P 0 = Probability (demand = 0) = 0.20 P 1 = Probability (demand = 1) = 0.25 P 2 = Probability (demand = 2) = 0.20 P 3 = Probability (demand = 3) = 0.25 P 4 = Probability (demand = 4) = 0.10 (i) Construct a payoff table for the given problem. (ii) Calculate the EMV for each decision. (iii) How many newspapers should the shopkeeper buy from the delivery truck? (1 mark) Page 3 of 10
(b) The research department of a company has recommended to the marketing department to launch a mobile phone of three different models. The marketing manager has to decide which one of the three models of the mobile phones is to be launched under the following estimated payoffs for various levels of sales. Estimated level of sales (units) ------------------------------------ Types of mobile Rs 15000 Rs 20000 Rs 25000 Nokia 45 25 12 Samsung 22 32 25 I Phone 32 38 20 What will be the manager s decision if the: (i) Maximax criterion (ii) Maximin criterion (iii) Laplace criterion (iv) Minimax regret criterion is applied in each case? (c) A company has a new product which is expected to have a good potential. At the moment the company has 2 courses of action open to the marketing department either test the market or drop the product. If the department tests the product, it will cost Rs 90,000 and the response could be positive or negative with the probabilities of 0.70 and 0.30 respectively. If it is positive, they would either market it with full scale, then the result might be low, medium or high demand and the respective net payoffs would be Rs 200,000, Rs 150,000 or Rs 400,000. These outcomes have probabilities of 0.35, 0.40 and 0.25 respectively. If the result of the test marketing is negative they have decided to drop the product. If, at any point, they drop the product there is a net gain of Rs 45,000 from the scale of scrap. All financial values have been discounted to the present. Draw a decision tree for the problem and indicate the most preferred decision. (6 marks) Page 4 of 10
SECTION B: ANSWER ANY THREE QUESTIONS QUESTION 2: (20 MARKS) A construction project consists of thirteen different activities denoted by the letters A, B,, M. The table below shows the duration of each activity and which activities if any must precede it. The project will be completed when activity M is completed. Activities Preceding Activities Duration (days) A - 2 B - 4 C - 1 D A 3 E A 7 F C 2 G C 4 H B, D 2 I B, F 6 J G 3 K E, H 2 L I 2 M K, L, J 1 (a) Construct a C.P.A.-network to represent this project, introducing dummy activities where necessary. (8 marks) (b) Say what is meant by the early event time ET (v) and the late time event LT (v) for an event v. Calculate ET (v) and LT (v) for each event in this project, entering these times in the appropriate boxes on your diagram. (c) What is the shortest time in which this project can be completed? (1 mark) Page 5 of 10
(d) Suppose an activity is represented by the arc v i v j with weight w (v i v j ). Explain how the total float time for this activity is calculated. Tabulate the total float time for each activity in this project. (e) Identify all the critical activities in this project and find the critical path(s) in this C.P.A.-network. QUESTION 3: (20 MARKS) The following linear programming problem models the problem of finding the optimal production schedule to maximize the profit on the manufacture of four related products, subject to an upper limit on the availability of three resources 1, 2 and 3.(represented by constraints (1), (2) and (3) respectively). Find x 1, x 2, x 3, x 4 є R to maximize z = 5x 1 + 2 x 2 + 4 x 3 + 3 x 4, subject to: 4 x 1 + 2 x 2 + x 3 + 2 x 4 150 (1) 6 x 1 + x 2 + 4 x 3 + 3 x 4 275 (2) 2 x 1 + x 2 + 3 x 3 + 2 x 4 100 (3) x 1, x 2, x 3, x 4 0. (4) (a) Prepare this problem for solution by the simplex algorithm and construct the initial tableau. (b) Find the first entering variable (EV) and the corresponding leaving variable (LV), giving a reason for your choice in each case. Page 6 of 10
(c) Complete the first iteration (only) of the simplex algorithm. Give the augmented basic feasible solution at the end of this iteration and an expression for z in terms of the current non-basic variables. Use this expression to explain why the current solution is not optimal. (5 marks) (d) After several iterations of the simplex algorithm, the following tableau is obtained, where x 5, x 6, x 7 are the slack variables in the constraints corresponding to resources 1, 2 and 3 respectively. Eq # z x 1 x 2 x 3 x 4 x 5 x 6 x 7 RS 0 1 0 0.5 0 0.6 0.7 0 1.1 215 1 0 1 0.5 0 0.4 0.3 0-0.1 35 2 0 0-2 0-1 -1 1-1 25 3 0 0 0 1 0.4-0.2 0 0.4 10 (i) (ii) Explain why the solution represented by this tableau is optimal. (1 mark) Give the values of the decision variables that give an optimal solution for this problem and the maximum value of z. (iii) Determine whether any resource is in surplus at the optimal solution, and by how much. (1 mark) (iv) Find the shadow prices for each resource. (1 mark) (v) Determine which resource you want to increase and by how much. (e) With the original allocation of resources, management are able to increase the unit profit on the first product from 5 to 6, so that the revised objective function becomes z 0 = 6x 1 + 2 x 2 + 4 x 3 + 3 x 4. Decide whether the solution represented by the tableau shown above is still optimal. Page 7 of 10
QUESTION 4: (20 MARKS) (a) A dairy firm has two plants located in a metropolitan city. Daily milk production at each plant is as follows: Plant 1-8 million litres Plant 2-12 million litres Plant 3-13 million litres Each day the firm must fulfill the needs of its three distribution centers. Minimum requirement at each centre is as follows: Distribution center 1-10 million litres Distribution center 2-12 million litres Distribution center 3-11 million litres. Cost of shipping one million litres of milk from each plant to each distribution centre is given in the following table in hundred of rupees: Shipping Cost Plant Distribution Centre 1 2 3 1 3 2 2 2 3 1 2 3 2 4 2 The dairy firm wishes to decide as to how much should be the shipment from which plant to which distribution centre so that the cost of shipment may be minimum. You are required to formulate this problem as a linear programming problem taking care in your answer to define the decision variables. (6 marks) Page 8 of 10
(b) Determine an initial feasible solution to the following transportation problem using: (i) the north-west corner rule; (ii) the row minima method. Origin Destination Availability D1 D2 D3 D4 O1 8 4 7 6 16 O2 5 6 3 9 17 O3 7 6 6 7 12 Requirement 9 11 15 10 (iii) Find also the transportation cost for each model. (c) Construct the dual of the following linear programming problem, P1. Find x 1, x 2, x 3 є R to minimize Z 1 = 2x 1 + 3 x 2-4 x 3, subject to: 2 x 1 - x 2 + x 3 4 x 1 + 2x 2-3 x 3 6 - x 1 + x 2 - x 3 2 x 1, x 2, x 3 0. (6 marks) Page 9 of 10
QUESTION 5: (20 MARKS) (a) What is meant by inventory? (b) Name the typical items carried in inventory. (c) What are the assumptions for the Economic Order Quantity (EOQ)? (d) What are the characteristics of a queuing system? (e) What are the implications of waiting lines? (f) List and explain briefly the most important factors to consider in analyzing a queuing system. (g) What are the assumptions of the basic simple queuing model? QUESTION 6: (20 MARKS) A study was conducted to determine the effects of sleep deprivation on subject s ability to solve simple problems. The amount of sleep deprivation varied from 8 to 24 hours and was carefully controlled. A total of 10 subjects participated in the study, 2 at each sleep deprivation level. After the specified period, each subject was given a set of simple addition problems and the number of errors was recorded. The data were No. errors, y 8 6 6 10 8 14 14 12 16 12 No. hours, x 8 12 16 20 24 Regarding y as a continuous variable, perform a linear regression analysis on these data: (a) Plot the data on a scatter diagram and make a brief comment on the scatter plot. (b) Postulate a regression model for these data mentioning carefully what assumptions are made on the random errors. (c) Find the corrected sum of squares and products. (d) Compute the least squares regression line. Add the line to your scatter plot. (e) Find the Pearson moment correlation coefficient, r and make a brief comment on the value you obtain. ***END OF QUESTION PAPER*** Page 10 of 10