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To find the optimum solution using MODI method, we need to first find the initial basic feasible solution using Vogel's approximation method. Let's write the initial transportation table:

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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

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

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.

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

34, 24 23 31 38 34 35 31 29 28 25 27 32 33 30

Calculate r₆, r₇, r₈ and r₉ and plot the correlogram

 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:

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

(b) Optimum solution using MODI method

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 

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 

State whether the following statements are True or False and also give the reason in support of your answer.
a) The Set $ ={(x, y) : 0 ≤ y ≤ 5 when 0 ≤ x≤ 2 and 3≤ y≤ 5 when 2≤ x ≤ 7 } is not a convex set.
b) 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.
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 di ≥ 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