Four professors are capable of teaching any one of four different courses. Class preparation time in hours for different topics varies from professor to professor and is given in the table below:
| Professor | A | B | C | D | ||||
| Linear Programming | 2 | 15 | 13 | 4 | ||||
| Queuing Theory | 10 | 4 | 14 | 15 | ||||
| Transportation Problem | 9 | 14 | 16 | 13 | ||||
| Regression Analysis | 7 | 8 | 11 | 9 | ||||
Each professor is assigned only one course. Determine an assignment schedule so as to minimise the total course preparation time for all courses.
This is an Assignment Problem, which can be solved using the Hungarian Method to minimize the total preparation time. ### Step 1: Construct the Cost Matrix Each professor (A, B, C, D) is assigned to one of the four courses, and the given preparation times are: | Course / Professor | A | B | C | D | |--------------------|----|----|----|----| | Linear Programming | 2 | 15 | 13 | 4 | | Queuing Theory | 10 | 4 | 14 | 15 | | Transportation Problem | 9 | 14 ________ ________ ______ ____ ______.
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The following table represents the sales (in thousands) of mobile sets of a shop for 16 quarters over four years:
| Year | Quarter | |||
| Q1 | Q2 | Q3 | Q4 | |
| 2011 | 554 | 590 | 616 | 653 |
| 2012 | 472 | 501 | 521 | 552 |
| 2013 | 501 | 531 | 553 | 595 |
| 2014 | 403 | 448 | 460 | 480 |
Using the graphical method to minimise the time required to process Job 1 and Job 2 on five machines A, B, C, D and E, find the minimum elapsed times an idle times to complete both jobs.
| Job 1 | Sequence | A | B | C | D | E | ||||
| Time (in hours) | 1 | 2 | 3 | 5 | 4 | |||||
| Job 2 | Sequence | C | A | D | E | B | ||||
| Time (in hours) | 3 | 4 | 2 | 1 | 5 | |||||
Use the penalty (Big M) method to solve the following LP problem:
Minimise Z 5x1+ 3x2
Subject to the constraints:
State whether the following statements are True or False. Give reasons in support of your answers.
(a) The solution of a transportation problem with 5 rows (supplies) and 4 columns (destinations) is feasible if number of possible allocations are 8.
(b) The moving averages of suitable period in a time-series are free from the influences of seasonal and cyclic variations.
(c) If the basic solutions for a system of equations are (-2, 0, 1), (0, 1, 3), (-2, 3, 0), then only (0, 1, 3) is feasible.
(d) In the stepwise selection method of multiple regression model, once a variable enters in the model then it always remains in the model.
(e) An enterprise requires 1000 units per month. The ordering cost is estimated to be 50 per order. The purchase price is 20 per unit and the carrying cost per unit is 10% of it. Then the economic lot size to be ordered is 775.
In a railway marshalling yard, goods trains arrive at a rate of 36 trains per day. Assuming that the inter-arrival and service time distributions both follow exponential distribution with an average of 30 minutes, calculate the following:
(i) Traffic intensity
(ii) The mean queue length
(iii) Probability that the queue size exceeds
A company has three production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100s) per week of a product, respectively. These units are to be shipped to four warehouses D1, D2, D3 and D4 with requirement of 5, 6, 7 and 14 units (in 100s) per week, respectively. The transportation costs (in Rs) per unit between factories to warehouses are given in the table below:
| Dl | D2 | D3 | D4 | Capacity | |
| S1 | 19 | 30 | 50 | 10 | 7 |
| S2 | 70 | 30 | 40 | 60 | 9 |
| S3 | 40 | 8 | 70 | 20 | 18 |
| Demand | 5 | 8 | 7 | 14 | 34 |
Seven successive observations on a stationary time-series are as follows:
12, 14, 13, 10, 15, 12, 15
(a) Calculate auto-covariances C0, C1, C2, C3 and C4.
(b) Calculate auto-correlation coefficients r1, r2, r3 and r4.
(c) Plot the correlogram.