Strategy	States of Nature			Minimum
	S1	S2	S3
d1	12	9	13	9
d2	15	11	8	8
d3	5	8	10	5

Max (9, 8, 5) = 9
∴ d₁ is the best strategy using maximin principle.

Alternative	Economy			Maximum
	Growing	Stable	Declining
Bonds	40	45	5	45
Stocks	70	30	-13	70
Mutual Funds	53	45	-5	53

Min (45, 70, 53) = 45
∴ Choosing Bonds is the best decision under minimax principle.

Action	States		Minimum
	B1	B2
A1	8	6	6
A2	9	2	2
A3	6	4	4

Max (6, 2, 4) = 6
∴ Action Al is the best according to maximin principle

Here total availability = 150 + 100 + 250 = 500
total requirement = 50 + 150 + 300 = 500
∴ Total availability = total requirement
∴ The given problem is a balanced transportation problem
Hence, there exists a feasible solution to the given problem
I - allocation:

[∵ least cost is 4 & min (50,150) = 50]
II - allocation:

[∵ least cost is 6 & min (150, 250) = 150]
III - allocation:

[∵ least cost is 8 & min (300, 100) = 100]
IV - allocation:

[∵ least cost is 9 & min (200,100) = 100]
V - allocation:

[∵ min (100, 100) = 100]
Thus, the allocations are

∴ The transportation schedule is
O₁ → D₁, O₁ → D₃, O₂ → D₃, O₃ → D₂, O₃ → D₃
Hence, the total transportation cost is
= 50(4) + 100(8) + 100(11) + 150(6) + 100(9)
= 200 + 800 + 1100 + 900 + 900
= Rs. 3900

Here, total supply = 10 + 5 + 3 = 18
total demand = 5 + 4 + 6 + 3 = 18
∴ Total supply = total demand
∴ The given problem is a balanced transportation problem.
∴ We can find an initial basic feasible solution to the given problem.
I - allocation:

[∵ min (5, 10) = 5]
II - allocation:

[∵ min (4, 5) = 4]
III - allocation:

[∵ min (1, 6) = 1]
IV - allocation:

[∵ min (5, 5) = 5]
V - allocation:

[∵ min (3, 3) = 3]
Thus, the allocations are

∴ The transportation schedule is
1 → A, 1 → B, 1 → C, 2 → C, 3 → D
Hence, the total transportation cost
= 5(3) + 4(1) + 1(7) + 5(5) + 3(2)
= 15 + 4 + 7 + 25 + 6 = Rs. 57

The assignment problem is a special case of transportation problem where the number of sources and destinations are equal. Here, jobs represent sources and machines represent destinations.

The mathematical form of assignment problem is minimize Z = \(\overset { n }{ \underset { i=1 }{ \Sigma } } \overset { n }{ \underset { j=1 }{ \Sigma } } { { C }_{ ij } }{ x }_{ ij }\) subject to the
Constraints \(\overset { n }{ \underset { i=1 }{ \Sigma } }{ x }_{ ij } = 1, j=1,2,.....n\)
\(\overset { n }{ \underset { j=1 }{ \Sigma } }{ x }_{ ij } = 1, i=1,2,.....n\) and x_ij = 0 or 1 for all i, j

To assign the different jobs to the different machines (one job per machine) to minimize the overall cost is known as assignment problem.

If the total supply = total demand, then the given problem is a balanced transportation problem.

A feasible solution to a transportation problem is a set of non negative values x_ij (i = 1, 2, m, j = 1, 2, ....... n) that satisfies the constraints.
If a basic feasible solution to a transportation problem contains exactly m + n - l allocations. in independent positions, it is called a non degenerate basic feasible solution. Here m is the number of rows and n is the number of columns in a transportation problem.

The objective function is minimize Z = \(\overset { m }{ \underset { i=1 }{ \Sigma } } \overset { n }{ \underset { j=1 }{ \Sigma } } { { C }_{ ij } }{ x }_{ ij }\) subject to the constraints
\(\overset { n }{ \underset { j=1 }{ \Sigma } }{ x }_{ ij } = a_i, i=1,2,.....m\) (Supply constraints)
\(\overset { m }{ \underset { i=1 }{ \Sigma } }{ x }_{ ij } = b_j, j=1,2,.....n\) (demand constraints)
x_ij ≥, 0 for all i, j (non-negative restrictions)

A transportation problem is to determine the amount to be transported from each origin to each destinations such that the total transportation cost is minimized.

Alternative	Pay – offs (Conditional events)				Minimum pay off
	A1	A2	A3	A4
E1	7	12	20	27	27
E2	10	9	10	25	25
E3	23	20	14	23	23
E4	32	24	21	17	32

min( 27, 25, 23, 32) = 23. Since the minimum cost is 23, the best alternative is E₃ according to minimax principle.

Action	States				Minimum
	(s1)	(s2)	(s3)	(s4)
A1	5	10	18	25	5
A2	8	7	8	23	7
A3	21	18	12	21	12
A4	30	22	19	15	15

Max (5,7,12,15) = 15 ஃ Action A4 is the best

First allocation :

Second allocation :

Third allocation :

Fourth allocation :

Fifth allocation :

The transportation cost is 
\( =(200 \times 11)+(50 \times 13)+(175 \times 18)+ (125 \times 14)+(150 \times 13)+(250 \times 10) \)
= 2200 + 650 + 3150 + 1750 + 650 + 2500 
= 10900