Directorate of distance education

Partial Derivatives
in finding a partial derivative, we hold all 
variables constant except one. Then we differentiate with respect to that 
one variable, treating all other variables as constants.
For example, if 
y = zx3-x+2x, 
let’s find the first partial derivative 
of 
y
with respect to 
x
. to do this, we treat 
z 
as though it were a constant 
and differentiate:
Optimizing the Simple Lot Size Formula 
The total cost equation for the simple lot size formula was developed 
to be:
Taking the first partial derivative of total cost with respect to order 
quantity, 
Q
: 
Setting the first partial derivative equal to zero, and solving for 
Q
: 
Taking the second partial derivative to ensure a minimum of the 
const function:

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Notes
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A positive value results, thus ensuring a minimum. Notice that 
the term 
CD, 
the total cost of the items, dropped out when we found the 
first derivative. This illustrates that this cost component is constant with 
regard to changes in order quantity.
Rationalisation
Again, we can see the power of the logic in calculus, but you need 
not be overwhelmed if you cannot follow all the mathematics. Clearly, the 
logic of mathematics is useful when applied to the many rational problems 
in productions / operations.
SCM Inventory Management
➢ Manufacturers would like to produce in large lot sizes because 
it is more cost effective to do so. The problem, however, is that 
producing in large lots does not allow for flexibility in terms of 
product mix.
➢ Retailers find benefits in ordering large lots such as quantity 
discounts and more than enough safety stock.
➢ The downside is that ordering/producing large lots can result in 
large inventories of products that are currently not in demand 
while being out of stock for items that are in demand.
➢ Ordering/producing in large lots can also increase the safety stock 
of suppliers and its corresponding carrying cost. It can also create 
what’s called the