Explanation Given: 2 x 1 + x 2 ≥ 20 2 x 1 + 5 x 2 ≤ 80 and z = 6 x 1 + 2 x 2 is maximized Explanation: Provided Maximize z = 6 x 1 + 2 x 2 Subject to: 2 x 1 + x 2 ≥ 20 (I) 2 x 1 + 5 x 2 ≤ 80 (II) Step 1: First, add slack variable and subtract surplus variable in order to get equality constraints. Subtract surplus variable (say s 1 ) from equation (I) to get 2 x 1 + x 2 − s 1 = 20 Add slack variable (say s 2 ) to equation (II) to get 2 x 1 + 5 x 2 + s 2 = 80 Step 2: Represent the system of linear equations obtained above in simplex tableau. This is the initial tableau: [ x 1 x 2 s 1 s 2 z s 1 2 1 − 1 0 0 20 s 2 2 5 0 1 0 80 z − 6 − 2 0 0 1 0 ] _ The solution from initial tableau is ( x 1 , x 2 , s 1 , s 2 ) = ( 0 , 0 , − 20 , 80 ) _ . This is not a feasible solution because the value of the surplus variable ( s 1 ) _ is negative. Step 3: If any basic variable has a negative value, identify the nonzero number in that variable’s column, and note that row. In the initial tableau, basic variable s 1 is negative and the nonzero number in this variable column is − 1 , which is in row 1, that is, in R 1 . Step 4: Now, in the row located in step 3, find the most positive entry in the left most column. The most positive entry in column 1 is 2 . Step 5: In the column found in step 4, pick a pivot by test ratio (or by quotients). Column 1 is a pivot column. Apply test ratio: 20 2 = 10 , 80 2 = 40 Therefore, 2 is the pivot because the smallest positive ratio is obtained by the element of R 1 , which follows that s 1 is the depart variable and x 1 is the entry variable. Therefore [ x 1 x 2 s 1 s 2 z x 1 2 1 − 1 0 0 20 s 2 2 5 0 1 0 80 z − 6 − 2 0 0 1 0 ] _ Since pivot is not 1 , make the pivot element 1 by R 1 → R 1 2

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

11th Edition

ISBN: 9780133886818

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Table 5: Measurement Data for Question 9 Part Number Op-1, M-1 Op-1, M-2 | Op-2, M-1 Op-2, M-2 | Op-3, M-1 Op-3, M-2 1 21 20 20 20 19 21 2 24 23 24 24 23 24 3 4 5 6 7 8 9 10 11 21 12 8222332 201 21 20 22 20 22 27 27 28 26 27 28 19 18 19 21 24 21 22 19 17 18 24 23 25 25 23 26 20 20 18 19 17 13 23 25 25 2 3 3 3 3 2 3 18 18 21 21 23 22 24 22 20 19 23 24 25 24 20 21 19 18 25 25 14 24 24 23 25 24 15 29 30 30 28 31 16 26 26 25 26 25 17 20 20 19 20 20 843882388 20 18 25 20 19 25 25 30 27 20 18 19 21 19 19 21 23 19 25 26 25 24 25 25 20 19 19 18 17 19 17 Question 9 A measurement systems experiment involving 20 parts, three operators (Op-1, Op-2, Op-3), and two measure- ments (M-1, M-2) per part is shown in Table 5. (a) Estimate the repeatability and reproducibility of the gauge. (b) What is the estimate of total gauge variability?" (c) If the product specifications are at LSL = 6 and USL 60, what can you say about gauge capability?

Question 5 A fraction nonconforming control chart with center line 0.10, UCL = 0.19, and LCL = 0.01 is used to control a process. (a) If three-sigma limits are used, find the sample size for the control charte 2 (b) Use the Poisson approximation to the binomial to find the probability of type I error. (c) Use the Poisson approximation to the binomial to find the probability of type II error if the process fraction defective is actually p = 0.20.

Question 8 Ten parts are measured three times by the same operator in a gauge capability study. The data are shown in Table 4. (a) Describe the measurement error that results from the use of this gauge. I (b) Estimate total variability and product variability. (c) What percentage of total variability is due to the gauge? (d) If specifications on the part are at 100±15, find the P/T ratio for this gauge. Comment on the adequacy of the gauge. 3 Table 4: Measurement Data for Question 8 Part Number Measurement 1 Measurement 2 Measurement 3 1 2 100 95 101 100 93 97 3 101 103 100 4 96 95 97 5 98 98 96 6 99 98 98 7889 95 97 98 100 99 98 100 100 97 10 100 98 98 99

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Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 2E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 4E Ch. 4.1 - For Exercises 5-8. (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - Introduce slack variables as necessary, then...Ch. 4.1 - Introduce slack variables as necessary, then write...

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