Inventory Control Guide Page 32
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404.5 Objective 3: Minimize the total costs
In the previous examples, R was computed to fulfill some service performance.
Decision:
Select Q and R = Lt × D + SS
Here a pure economical analysis is carried on.
Objective 3
Minimize the total costs assuming penalty costs
for the demand which cannot be served from stock.
Here we assume that each unit which is demanded and which cannot be immediately served
generates a cost P.
Penalty Costs: P
(money/unit)
lost profit / contractual penalty / loss of goodwill
The determination of the penalty cost is difficult in practice. Here are some examples. If you
sell newspapers on the street, and if you run out of stock, for each unit you do not sell, you
loose the corresponding profit. If you are a baker and you run out of breads. Not only you
loose the profit associated with each bread, but you could also loose a customer.
Minimize
C(Q, R) = O (D/Q)+ H (Q/2+R-D Lt)
+ ID +P n(R) (D/Q)
In the above expression, the number of cycles has been set to D/Q. This means that a
demand equal to Q is served during each cycle. This is thus a backorder model.
By computing the partial derivatives and setting them to zero, one obtains
+
2D[O P n( )]
R
=
Q
H
∂
n( )
R
Q
H
−
=
>
=
Pr[D
R
]
Lt
∂
R
P D
which should be solved iteratively.
Method 3. If the penalty cost P is known, then
=
+
Q
2D[O P n( )] / H
R
solve iteratively
>
=
Pr[D
R
]
Q
H / P D
Lt
Let us try to understand both equations intuitively. First, the penalty cost per cycle Pn(R) can
be seen as an additional fixed cost per cycle. It does indeed play the same role as O does.
This explains the equation for Q. Let us now consider the second equation. If you decide to
increase your safety stock by one unit, it will cost you an additional holding cost of HQ/D per
cycle. On the other hand, when you can sell this unit that is when the demand exceed R, you
save the penalty cost associated to this unit. This is the term P Prob[D(Lt) >R].
You should thus select the order point which equilibrates both costs.
Prod 2100-2110
Inventory Control
31
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