Cork Regional Technical College

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1 Cork Regional Technical College Bachelor of Engineering in Chemical and Process Engineering - Stage 4 Summer 1996 CE DECISION ANALYSIS AND OPERATIONS RESEARCH (Time: 3 Hours) Answer FOUR Questions. All Questions carry equal marks. Statistical tables are available. Examiners: Dr. D. Menzies Prof. J. Garside Mr. D. O'Hare 1. (a) A refinery blends four petroleum components into three grades of petrol - regular, premium and unleaded. The maximum quantities available of each component and the cost per barrel are as follows: Component Maximum Barrels Available/Day Cost (per barrel 1 5, , , ,500 9 To ensure that each grade of petrol retains certain essential characteristics, the refinery has put limits on the percentages of the components in each blend. The limits as well as the selling prices for the various grades are as follows:

2 Grade Component Specifications Selling Price (barrel) Regular Not less than 40% of 1 16 Not more than 20% of 2 Not less than 30% of 3 Premium Not less than 40% of 3 22 Unleaded Not more than 50% of 2 14 Not less than 10% of 1 The refinery wants to produce at least 3,000 barrels of each grade. Management wishes to determine the optimal mix of the four components that will maximise profit. Formulate, but do not attempt to solve, a linear programming model for this problem. (Hint: Use 12 decision variables). Solve the L.P. problem minimise Z = 8x x 2 + 4x 3 subject to x 1 + x 2 + x x 1 + x 2 60 x 1, x 2, x 3 0 (c) Write down the dual of the problem in. Use the optimal table in to determine the optimal values of the dual variables and objective function. Verify the solution in by solving the dual problem graphically. 2. (a) Use the two-phase method to show that there is no feasible solution to the following set of constraints: 2x 1 + 4x 2 + x 3 8 x 1 + 3x 2 + x 3 9 x 1, x 2, x 3 0. Explain the term SENSITIVITY ANALYSIS. 2

3 (c) A manufacturing firm produces electric motors for washing machines and vacuum cleaners. The firm has resource constraints for production time, steel and wire. The linear programming model for determining the number of washing machine motors (x 1 ) and vacuum cleaner motors (x 2 ) to product has been formulated as follows. maximise Z = 70x x 2 (profit, ) subject to 2x 1 + x 2 19 (production, hr.) x 1 + x 2 14 (steel, kg) x 1 + 2x 2 20 (wire, m.) x 1, x 2 0 The optimal table is as follows: Basis Z x 1 x 2 S 1 S 2 S 3 Solution x / / 3 6 S / / 3 1 x / / 3 7 (Z) Interpret the optimal table. It is proposed to increase the amount of time available in the production department. By how much may the time be increased before the current basis becomes infeasible? How much is it worth paying for this increase? A further constraint, relating to administration must now be taken into consideration. This constraint is x 1 + x 2 10 What are the implications of this constraint for the current solution? If necessary, find a new optimal point. (iv) The firm is considering producing a third type of motor. Each such motor would require 1 hour of production time, 1 kg of steel and 3 m of wire. The profit per unit would be 120. What now is the optimal solution? 3

4 3. (a) Explain the terms dummy source penalty cost as they apply in the solution of transportation problems. A firm producing a single product has three plants and four customers. The three plants will produce 6, 8 and 4 units respectively, during the next time period. The firm has made a commitment to sell 4 units to customer 1, 6 units to customer 2 and at least 2 units to customer 3. Both customers 3 and 4 also want to buy as many of the remaining units as possible. The net profit associated with shipping 1 unit from plant i to customer j is given by the following table: Customer Plant Management wishes to know how many units to sell to customers 3 and 4 and how many units to ship from each of the plants to each of the customers to maximise profit. Formulate this problem as a transportation problem and find the optimal solution. (c) The following is the effectiveness matrix in an assignment problem: Job Individual J K L M N A B C D E Use the Hungarian method to find the optimal assignment, given that the problem is a maximisation problem. 4

5 4. (a) Explain the terms dummy activity crashing the network as they occur in Critical Path Analysis. The following table provides information on a research project. Expected Time Direct Cost Activity Predecessor Regular Crash Regular Crash a b c a d c e b, c Using regular times, find the critical path. State the expected time and cost. If a saving in overhead costs of 150 per day may be made by reducing the overall project completion time, find the optimum project completion time from the point of view of overall cost. (c) The following are details on the activities in a small project. Completion Times (days) Activity Predecessor Optimistic Most Likely Pessimistic a b c a d b, c e a Find the expected value and variance of the completion time for each activity. Use the expected times in to find the critical path. Assuming the normal distribution applies, determine the probability that the critical path will take between 18 and 26 days to complete. 5

6 5. (a) In a pharmaceutical firm, the variation in the weight of an antibiotic from batch to batch is important. With the present process, the standard deviation is 0.12 g. The research department has developed a new process which they believe will result in less variation. The following weight measurements (in grams) were obtained with the new process: 6.47, 6.49, 6.64, 6.59, 6.55, 6.52, Use both a Χ 2 test and an F-test to determine if the data provide significant evidence to support the researchers' hypothesis. State your conclusions clearly. In a one factor analysis of variance with 5 treatment groups the sample sizes are n 1 = 7, n 2 = 6, n 3 = 8, n 4 = 6, n 5 = 8. How many degrees of freedom are there in the residual variance? Explain your answer. An experiment was performed to determine the effect of four different chemicals on the strength of a fabric. Five fabric samples were selected and a randomised block design was run by testing each chemical type once in random order on each fabric sample. The data are shown below. Fabric Samp le Chemical Type Complete the appropriate ANOVA table, carry out the test of hypothesis and comment. (c) A process engineer is trying to improve the life of a cutting tool. He has run a 2 3 experiment using cutting speed (A), metal hardness (B) and cutting angle (C) as the factors. The data obtained are as follows: Run (1) a b ab c ac bc abc Response

7 Use Yate's algorithm to analyse the data and state your conclusions. 6. (a) A decrease in sample size when using an x R chart will do which of the following: I II III widen 3 σ limits reduce process variability increase sensitivity. Choose an answer from: (iv) I only II only II and III only I, II and III. Samples of four units were taken from a manufacturing process at regular intervals. The width of a slot on a part was measured and the average and range were computed for each sample. After 25 samples of four, the following values were obtained: x = 614 R = 16.5 The data were plotted on an X R chart and all points fell within the control limits (3 σ limits). What are those control limits? Calculate the capability indices C p and C pk. What percentage of product would you expect to fall outside specifications, if the specifications were ? (c) A control chart for averages and ranges has been used to help control a manufacturing process. The sample data are consistently within control limits and the control limits are inside the engineering tolerance limits. The supervisor is confused because a high percentage of product is outside the tolerance limits even though the process is in control. How would you account for this? (d) A p-chart, using 3σ limits, has centre line at 0.02, UCL at 0.05 and LCL at 0. What sample size is being used? Suppose the process fraction defective shifts to What is the probability of detecting this shift on the first subsequent sample? by no later than the fourth subsequent sample. 7

8 (e) Suppose that a manufacturer ships batches of size A single sampling plan with n = 50 and c = 2 is being used for receiving inspection. Rejected batches are screened and all defective items are reworked and returned to the batch. What level of batch quality will be rejected 5% of the time? 95% of the time? If incoming batches are 1% non-conforming, calculate the ATI (average total inspection per batch). 8