LECTURE ON 16/10/2002

 

Continuing from the last lecture in which we had considered random demand which is continuously distributed with a PDF (Probability Distribution Function) f(.).

The demand was represented as D and the order quantity as Q

It was derived in the last lecture that the cost for an order Q and demand D is given as

 

C(Q,D) = C_o (Q-D)⁺ + C_u (D-Q)⁺

 

Where C_o is the cost of over ordering

             C_u is the cost of under ordering (loss in profit)

            (x)⁺ = x if x>= 0

                 =  0 otherwise

 

Expected value of C(Q,D) is given by

 

 

For discrete probability distribution

 

 

 

C(Q) is of the type

 

 

 

 

 

 

 

 

 

 

 

For minimum cost C’(Q) = 0 therefore

 

Using Leibniz’s Rule which is

 

 

 

Where F(Q*) is called the CDF (Cumulative Distribution Function) of the demand. In other words it is the probability of demand being less than or equal to Q*.

 

 

Therefore C’’(Q*) > 0, hence it is a minima.

 

It can also be said that if we order Q* the probability that the complete demand gets satisfied is C_u/(C_u+C_o)

 

In order to find out the distribution of demand we can study the past data. Hypothesize a distribution and check it using a Goodness of Fit test (like c² test). However, there is an inherent problem in this as the past data only gives the amount ordered not the unmet demand.

 

For a problem in which

 

C – cost price

h – inventory cost per unit

S – sales price per unit

 

To find the minima of cost we add

As this a constant and adding a constant does not change the optimal solution there is no change in the solution. The equation reduces to

It can now be said that C_o = C+h and C_u = S-C (loss in profit)

 

The optimal is given by F(Q*) = C_u/(C_u+C_o) = (S-C)/(S+h)

 

Given an initial inventory “x” and there is no ordering cost

We order (Q*-x) if Q*>x and no order is placed otherwise

 

 

 

 

 

 

If there is an ordering cost “k” involved

 

 

 

 

 

 

 

 

 

 

 

 

If initial inventory is less than s order else don’t order as in that case ordering cost will exceed the benefit.

 

In general this is known as the (s,S) Policy or the two bin policy i.e. have a safety stock  with capacity s and a main stock with capacity S-s. Consume main stock first and order when it is finished and order such that both the stock/bins get filled. The probability of safety stock being consumed is low but inventory cost is also not very high.

 

Example – Consider a bicycle whose production is being discontinued and hence no more reorders are possible. Find the optimal ordering if

 

Cost price, C – Rs. 2000

Set up cost for ordering – 0

Cost of maintenance and inventory – Rs 100

Salvage value – Rs 1000

Selling price, S – Rs 4500

Zero inventory initially

Demand is exponential with mean – 10000 units

 

There fore the PDF of demand is given by

 

       = 0 otherwise

 

C_o = Cost incurred – salvaged

     = 2000 + 100 – 1000

     = 1100

C_u = Loss in profit

     = S – C

     = 2500

 

Therefore for optimal ordering Q*, F(Q*) = C_u/(C_u+C_o) = 0.6944

 

 

 

 

Þ Q* = 11856

 

Now if we add an ordering cost k = Rs 80,000

 

We have to find both s and S

S = Q* = 11856

For finding s we use the equation

C(s) = k + C(S)

 

Where

Where

f(x) = le^-lx for x>= 0

      = 0 otherwise

 

l = 1/10000, S = 11856

 

 

Similarly the other integral can also be evaluated

 

C(S) = 2500*3055.63 + 1100*4911.63 = 13041866

 

C(s) = k + C(S) = 80000+13041866

        = 13121866

 

s can be calculated numerically by solving the above equation to get s = 10,674 for which C(s) = 13121854.

 

Hence if the initial inventory is less than 10,674 it is better to order otherwise not.