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1. State whether the following statements are True or False and also give the reason in support of your answer.

a) The Set \(\S=\{(x, y): 0 \leq y \leq 5\) when \(0 \leq x \leq 2\) and \(3 \leq y \leq 5\) when \(2 \leq x \leq 7\}\) is not a convex set.

b) If 10 is added to each of the entries of the cost matrix of a \(3 \times 3\) assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10 .

c) The solution to a transportation problem with m-rows (supplies) and n-columns (destinations) is feasible if number of positive allocations is \(m+n\).

d) The Value \(d_{i} \geq 3\) indicates an outlying observation in regression analysis.

e) Variations which occur due to natural forces and operate in a regular and periodic manner over a span of less than or equal to one year are termed as cyclic variations.

2. (a) Rewrite the following linear programming problem in Standard form:

Minimise \(\quad Z=2 x_{1}+x_{2}+4 x_{3}\)

Subject to the Constraints:

\[
\begin{aligned}
& -2 x_{1}+4 x_{2} \leq 4 \\
& x_{1}+2 x_{2}+x_{3} \geq 5 \\
& 2 x_{1}+3 x_{3} \leq 2 \\
& x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0
\end{aligned}
\]

(b) Solve the following LPP using graphical method:

Maximize \(\quad \mathrm{Z}=3 \mathrm{x}_{1}+2 \mathrm{x}_{2}\)

Subject to the Constraints:

\[
\begin{gathered}
-2 x_{1}+x_{2} \leq 1 \\
x_{1} \leq 2 \\
x_{1}+x_{2} \leq 3 \\
x_{1}, x_{2} \geq 0
\end{gathered}
\]

3. Solve the following LPP using Simplex method:

Maximize \(\mathrm{Z}=\mathrm{x}_{1}+2 \mathrm{x}_{2}\)

Subject to the Constraints: \(-\mathrm{x}_{1}+2 \mathrm{x}_{2} \leq 8\)

\[
\begin{aligned}
& x_{1}+2 x_{2} \leq 12 \\
& x_{1}-x_{2} \leq 3 \\
& x_{1}, x_{2} \geq 0
\end{aligned}
\]

4. 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:

How should the tasks be allocated, one to a subordinate, so as to minimise the total man hour?

5. a) Use graphical method to minimise the time added to process the following jobs on the machines shown:

\(\begin{array}{lllllll}\text { Job 1: } & \text { Sequence } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ & \text { Time } & 3 & 4 & 2 & 6 & 2 \\ \text { Job 2: } & \text { Sequence } & \text { B } & \text { C } & \text { A } & \text { D } & \text { E } \\ & \text { Time } & 5 & 4 & 3 & 2 & 6\end{array}\)

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):

Calculate the residuals and determine the standardised residuals for the model \(\mathrm{Y}=2.6185+0.4369 \mathrm{X}\)

6. a) A Statistician collected the data of 78 values with two independent variable \(X_{1}\) and \(X_{2}\), and considered the four models:

(i) \(\mathrm{Y}=\mathrm{B}_{0}+\mathrm{e}\); (ii) \(\mathrm{Y}=\mathrm{B}_{0}+\mathrm{B}_{1} \mathrm{X}_{1}+\) e; (iii) \(\mathrm{Y}=\mathrm{B}_{0}+\mathrm{B}_{1} \mathrm{X}_{1}+\mathrm{e}\) and

(iv) \(Y=B_{0}+B_{1} X_{1}+B_{2} X_{2}+e\).

The results obtained are: \(\hat{\sigma}^{2}=0.91, \mathrm{SS}\left(\mathrm{B}_{0}\right)=652.42, \mathrm{SS}\left(\mathrm{B}_{0}, \mathrm{~B}_{1}\right)=679.34\), \(\mathrm{SS}\left(\mathrm{B}_{0}, \mathrm{~B}_{2}\right)=654.00\), and \(\mathrm{SS}\left(\mathrm{B}_{0}, \mathrm{~B}_{1}, \mathrm{~B}_{2}\right)=687.79\). Find the additional contribution of (i) \(X_{2}\) over \(X_{1}\) and (ii) \(X_{1}\) over \(X_{2}\). Test whether their inclusion in the model is justified.

b) Fifteen successive observations on a stationary time series are as follows:

\(\begin{array}{llllllllllll}34, & 24 & 23 & 31 & 38 & 34 & 35 & 31 & 29 & 28 & 25 & 27\end{array}\)

\(\begin{array}{llll}32 & 33 & 30\end{array}\)

Calculate \(\mathrm{r}_{6}, \mathrm{r}_{7}, \mathrm{r}_{8}\) and \(\mathrm{r}_{9}\) and plot the correlogram.

7. Calculate seasonal indices by the ratio to moving average method from the following data:

8. Consider the following Transportation problem:

It is not possible to transport any quantity from Factory B to Godown 5. Determine:

(a) Basic Feasible Solution by Vogel’s Approximation Method.

(b) Optimum solution using MODI method.