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Statistics

Notes Alternative Method

The work of computations of EMV's, in the above example, can be reduced considerably by the
use of the concept of expected marginal profit. Let p be the marginal profit and l be the marginal
loss of ordering an additional unit of the product. Then, the expected marginal profit of ordering
the Ath unit, is given by

= p .P (D ³ A ) l- .P (D < A ) p= .P (D ³ A ) l- [ . 1 P- (D ³ A )]

.P
= (p + l ) (D ³ A ) l- .... (1)
The computations of EMV, for alternative possible values of A, are shown in the following table:
In our example, p = 3 and l = 2 .

Thus, the expression for the expected marginal profit of the Ath unit
a a a
= 3 +2fP D ³ Af- 2 = 5P D ³ Af- 2.
Table for computations

Total profit or
Action ( ) P (D ³ A ) * EMP = 5P (D ³ A ) 2-
A
EMV
5 1.00 5 1.00 2 = 3.00 5 3.00 = 15.00
-
´
´
-
6 0.90 5 0.90 2 = 2.50 15.00 2.50 = 17.50
´
+
-
´
7 0.70 5 0.70 2 = 1.50 17.50 1.50 = 19.00
+
8 0.40 5 0.40 2 = 0.00 19.00 0.00 = 19.00
-
+
´
-
-
´
9 0.15 5 0.15 2 = - 1.25 19.00 1.25 = 17.75
* This column represents the 'more than type' cumulative probabilities.
Since the expected marginal profit (EMP) of the 8th unit is zero, therefore, optimal order is 8
units.
8.1.2 Marginal Analysis
Marginal analysis is used when the number of states of nature is considerably large. Using this
analysis, it is possible to locate the optimal course of action without the computation of EMV's
of various actions.
An order of A units is said to be optimal if the expected marginal profit of the Ath unit is non-
negative and the expected marginal profit of the (A + 1)th unit is negative. Using equation (1),
we can write
P
(p + l ) (D ³ A ) l- ³ 0 and .... (2)
P
(p + l ) (D ³ A + ) 1 - l < 0 .... (3)
From equation (2), we get

l l
P (D ³ A ) ³ or 1 P- (D < A )³
p + l p + l
l p
or P (D < A ) 1£ - or P (D £ A - ) 1 £ .... (4)
p + l p + l
[P(D £ A – 1) = P(D < A), since A is an integer]

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