# OR-Notes

## J E Beasley

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

#### Inventory control tutorial solution

For this problem:

R=150,000 (assuming constant selling rate) co=50 ch=0.2x5 (direct storage cost) + 0.15x5 (money tied up) = 1.75

Using the package we find that the EOQ is 2927.7 (i.e. 2928) with the associated total inventory cost being £5123.5. This implies that we are ordering R/2928 = 150000/2928 times a year i.e. 51.2 times a year.

Given that this is virtually once a week we might as well order exactly once a week (i.e. order R/52 = 150000/52 = 2885 at each order). Using the package the cost associated with this order quantity is £5124 - so, as we would expect, virtually the same as the cost associated with the EOQ.

Currently we are ordering 5 times a year, so the order quantity is R/5 = 30,000 at each order. Using the package we see that with this order quantity the associated total inventory cost is £26500. Hence by adopting a more frequent ordering policy (ordering once a week) our cost saving is £(26500 - 5124) = £21376.

Note here that we could have added the total material (purchase) cost of £750,000 to the above inventory costs if we had wished.

For this problem:

R=10,000 (assuming constant selling rate) co=10 ch=0.15x2 (direct storage cost) + 0.05x2 (money tied up) = 0.4

• EOQ is 707
• optimal decision to order 756 for 12.5% discount (probably then order 833 monthly) - this assumes holding cost is also discounted
• with new discount structure order 1000 for 25% discount

For this problem:

R=105,000 (assuming constant selling rate) co=100 ch=0.2x0.15 (direct storage cost) + 0.15x0.1 (money tied up) = 0.045

This is a problem with quantity discounts. Using the package we find that the optimal decision is to order 22771 with the associated total (inventory plus material) cost being £15097.2.

If there is a lead time of two weeks between placing an order and delivery of the order then the optimal decisions (and the cost) remain unchanged. What does change however is the point at which we order. The traditional inventory assumption is that orders are received instantaneously so that the reorder point is zero (wait until the stock level falls to zero before ordering) With a lead time we need to order before the stock level falls to zero and for a two week lead time we order 2 weeks before the stock level falls to zero. From the package we have that the reorder point with a lead time of two weeks is a stock level of 4038 (=2(R/52)).

With a replenishment rate of 25,000 units a month (i.e. 300,000 units a year) and a lead time of 2 weeks using the package tells us that the optimal decision is now to order 28243.95 (i.e. 28244) with an associated total (inventory plus material) cost of £14918.5. Note that this is less than before (£15097.2 before) as the inventory cost is less - replenishing stock gradually (rather than instantaneously) results in a lower average inventory level and hence a lower total inventory cost. This can been seen from the package output - the maximum inventory level is 22771 (the order quantity) with instantaneous replenishment but only 18359 with a replenishment rate of 25,000 a month. The moral of this is that it may be worth your while to have suppliers deliver gradually!