Reference no: EM132314914
Industrial Statistics Assignment -
Note: All questions are compulsory. Answer in your own words.
Q1. State whether the following statements are true or false and also give the reason in support of your answer.
(i) If 10 is added to each of the entries of the cost matrix of a 3 x 3 assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10.
(ii) The solution to a transportation problem with m-rows (supplies) and n-columns (destinations) is feasible if number of positive allocations is m + n.
(iii) If the arrival rate is 6 per hour and service rate is 2 per hour, then the probability of no customer in queue is 0.7.
(iv) A time series is a set of values arranged in geographical order.
(v) If the coefficient of determination is 0.933, the number of observations and independent variables are 10 and 2, respectively, then Adjusted R2 will be 0.84.
Q2. (a) Rewrite the following linear programming problem in Standard form:
Minimise Z = 2x1 + x2 + 4x3
Subject to the Constraints:
-2x1 + 4x2 ≤ 4
x1 +2x2 + x3 ≥ 5
2x1 + 3x3 ≤ 2
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
(b) Solve the following LPP using graphical method:
Maximize Z = 3x1 + 2x2
Subject to the Constraints:
-2x1 + x2 ≤ 1
x1 ≤ 2
x1 + x2 ≤ 3
x1, x2 ≥ 0
Q3. A department head has four subordinates, and four tasks to be performed. The subordinates differ in efficiency, and the tasks differ in their intrinsic difficulty. His estimate, of the time each man would take to perform each task, is given in the table below:
|
Tasks
|
Subordinates
|
|
E
|
F
|
G
|
H
|
|
A
|
18
|
26
|
17
|
11
|
|
B
|
13
|
28
|
14
|
26
|
|
C
|
38
|
19
|
18
|
15
|
|
D
|
19
|
26
|
24
|
10
|
How should the tasks be allocated, one to a subordinate, so as to minimise the total man hour?
Q4. a) Use graphical method to minimise the time added to process the following jobs on the machines shown:
|
Job 1
|
Sequence
|
A
|
B
|
C
|
D
|
E
|
|
Time
|
3
|
4
|
2
|
6
|
2
|
|
Job 2
|
Sequence
|
B
|
C
|
A
|
D
|
E
|
|
Time
|
5
|
4
|
3
|
2
|
6
|
Calculate the total time elapsed to complete both the jobs.
b) The following data comprising the number of customers (in hundred) and monthly sales (in thousand Rupees):
|
Number of Customers (in hundred)
|
4
|
6
|
6
|
8
|
10
|
14
|
18
|
20
|
22
|
26
|
28
|
30
|
|
Monthly Sales (in thousand Rs)
|
1.8
|
3.5
|
5.8
|
7.8
|
8.7
|
9.8
|
10.7
|
11.5
|
12.9
|
13.6
|
14.2
|
15
|
Calculate the residuals and determine the standardised residuals for the model Y = 2.6185 + 0.4369 X.
Q5. (a) For the following series of observations, verify that the 4-year centered moving average is equivalent to a 5-year weighted moving average with weights 1, 2, 2, 2, 1, respectively:
|
Year
|
2003
|
2004
|
2005
|
2006
|
2007
|
2008
|
2009
|
2010
|
2011
|
2012
|
2013
|
|
Annual Sales (in 000 Rs.)
|
20
|
60
|
10
|
50
|
30
|
70
|
20
|
60
|
40
|
80
|
30
|
(b) A statistician collected data of 50 values with two independent variables X1 and X2, in the process of fitting the following models (i) Y = B0 + e (ii) B0 = B1X1 + e (iii) Y = B0 + B2X2 = e and (iv) Y = B0 + B1X1 + B2X2 + e. The following results obtained: B^0 = 52.38, B^1 = 31.6161, B^2 = 0.0414, SS(B0) = 125.26, SS(B0;B1) = 179.08, SS(B0, B2) = 171.25, SS(B0, B1, B2) = 180.79 and σ^2 = 0.91.
Apply all the three selection procedures to choose an appropriate model.
Q6. Customers arrive at a one-man barber shop according to the Poisson process with a mean inter arrival time of 12 minute. Customers spend an average of 10 minute. in the barber's chair.
i) What is the expected number of customers in the barber shop and in the queue?
ii) Calculate the percentage of time an arrival can walk straight into the barber's chair without having to wait.
iii) How much time can a customer expect to spend in the barber's shop?
iv) Management will provide another chair and hire another barber, when a customer's waiting time in the shop exceeds 1.25hour. How much must the average rate of arrivals increase to warrant a second barber?
v) What is the average time customers spend in the queue?
vi) Calculate the percentage of customers who have to wait prior to getting into the barber's chair.
vii) What is the probability that more than 3 customers are in the system?
Q7. a) Calculate seasonal indices by the ratio to moving average method from the following data:
|
Year Quarter
|
2001
|
2002
|
2003
|
2004
|
|
Q1
|
750
|
860
|
900
|
1000
|
|
Q2
|
600
|
650
|
720
|
780
|
|
Q3
|
540
|
630
|
660
|
720
|
|
Q4
|
590
|
800
|
850
|
930
|
b) For the following Auto regressive model Xt = 0.6Xt-1 - 0.3Xt-2 + at
i. Verify whether the series is Stationary.
ii. Obtain ρk: k = 1, 2, 3, 4 and 5.
iii. Plot the Correlogram.
Q8. Consider the following Transportation problem:
|
Factory
|
Godowns
|
Stock Available
|
|
1
|
2
|
3
|
4
|
5
|
6
|
|
A
|
7
|
5
|
7
|
7
|
5
|
3
|
60
|
|
B
|
9
|
11
|
6
|
11
|
-
|
5
|
20
|
|
C
|
11
|
10
|
6
|
2
|
2
|
8
|
90
|
|
D
|
9
|
10
|
9
|
6
|
9
|
12
|
50
|
|
Demand
|
60
|
20
|
40
|
20
|
40
|
40
|
|
It is not possible to transport any quantity from Factory B to Godown 5. Determine basic Feasible Solution by Vogel's Approximation Method and optimum solution using MODI method.