Method and computer system for setting inventory control levels from demand inter-arrival time, demand size statistics

1. A method of determining minimum (s) and maximum (S) inventory control levels (s, S) for inventory items based upon a demand history of the inventory items, comprising:

using a computer to execute:

generating histograms of demand inter-arrival times (IA) and demand requisition sizes (RS), based upon the demand history of the inventory items;

generating, for a set of control levels (s, S), a probability distribution for a lead time demand (LTD) in future, based upon the IA and/or RS histograms, and a probability distribution for an inventory position (IP) based upon the RS histogram;

generating a probability distribution of on-hand inventory of an inventory item, for a lead time in future, directly from frequency statistics of different demand IA and demand RS on an inventory item basis based upon the probability distribution for the LTD and the IP;

computing a cost function for an expected instantaneous cost of a given IP of the inventory item, experienced for the lead time in future, based upon the probability distribution of the on-hand inventory of the inventory item; and

searching for the control levels (s, S) of the inventory item using the cost function or the expected instantaneous cost of the current IP of the inventory item.

2. The method according to claim 1 , wherein the expected instantaneous cost of the given IP of the inventory item experienced for the lead time in future includes one or more of an instantaneous cost of outstanding backorders and an instantaneous cost of carrying inventory.

3. The method according to claim 2 , wherein the instantaneous cost of carrying inventory includes expected on-hand inventory conditioned on a current IP and/or expected full IP, and the instantaneous cost of outstanding backorders includes unit backorders and requisition backorders.

4. The method according to claim 2 , further comprising:

averaging the instantaneous costs of outstanding backorders and carrying inventory across the probability distribution for the IP; and

applying a unimodal approximation to the instantaneous costs of outstanding backorders and carrying inventory that meets one or more conditions for searching.

5. The method according to claim 4 , wherein a total expected instantaneous cost of the inventory item experienced for the lead time in future includes a result of the averaging of the instantaneous costs of outstanding backorders and carrying inventory, and

wherein a total expected cost includes the total expected instantaneous cost and expected inventory item replenishment cost.

6. The method according to claim 1 , wherein the generating of the histograms comprises weighting the demand history of the inventory items by an adjustment factor that decreases as amount of time since data was collected increases.

7. The method according to claim 1 , wherein the generating of the RS histogram comprises resealing down requisition sizes of items with large requisition sizes by a virtual unit-of-issue factor and after the searching of the inventory item control levels (s, S), resealing up by multiplying each of s and S by the virtual unit-of-issue factor.

8. A method of determining minimum (s) and maximum (S) inventory control levels (s, S) for inventory items based upon a demand history of the inventory items, comprising:

using a computer to execute:

generating a probability distribution for a lead time demand (LTD) in future and a probability distribution for an inventory position (IP), based upon demand inter-arrival times (IA) and demand requisition sizes (RS) of the inventory items from the demand history of the inventory items; and

computing an expected cost including two or more inventory item cost components of expected cost of ordering, expected cost of carrying inventory, or expected cost of outstanding backorders for the inventory control levels (s, S), directly from frequency statistics of different demand IA and demand RS on an inventory item basis; and

searching for inventory control levels (s, S) of each inventory item that minimizes the cost of ordering, carrying inventory, and/or backorders, based upon the computed expected cost.

9. The method according to claim 8 , wherein

the computing of expected cost of outstanding backorders includes computing expected cost of unit backorders and/or computing expected cost of requisition backorders, and

the computing of expected cost of carrying inventory includes computing expected cost of on-hand inventory and/or computing expected cost of on-hand inventory and expected cost of full inventory.

10. The method according to claim 8 , wherein the generating of the LTD and the IP probability distributions is based upon generating of histograms of the demand IA and the demand RS from the demand history of the inventory items.

11. The method according to claim 10 , wherein the generating of the LTD and/or the IP probability distributions include a renewal process of one or more of a renewal function, a renewal density, or renewal-reward function.

12. The method according to claim 8 , wherein the searching includes searching for a subset of inventory control levels (s, S) according to Zheng-Federgruen optimization.

13. The method according to claim 10 , wherein the generating of the demand IA and RS histograms is without assuming probability distributions, and/or by not estimating mean and/or variance parameters for the probability distributions, thereby the computing of the expected cost is without theoretical probability distributions.

14. The method according to claim 8 , wherein the searching for the inventory control levels (s, S) optimizes investment across inventory items to provide lowest total backorders and/or total replenishment cost.

15. An apparatus, comprising:

one or more computing processors configured to execute:

accessing a demand history of inventory items;

generating a probability distribution for a lead time demand (LTD) in future and a probability distribution for an inventory position (IP), based upon demand inter-arrival times (IA) and demand requisition sizes (RS) of the inventory items from the demand history of the inventory items; and

computing an expected cost including two or more inventory item cost components of expected cost of ordering, expected cost of carrying inventory, or expected cost of outstanding backorders for the inventory control levels (s, S), directly from frequency statistics of different demand IA and demand RS on an inventory item basis; and

searching for minimum (s) and maximum (S) inventory control levels (s,S) of each inventory item that minimizes the cost of ordering, carrying inventory, and/or backorders, based upon the computed expected cost.

16. The apparatus according to claim 15 , wherein

the computing of expected cost of outstanding backorders includes computing expected cost of unit backorders and/or computing expected cost of requisition backorders, and

the computing of expected cost of carrying inventory includes computing expected cost of on-hand inventory and/or computing expected cost of on-hand inventory and expected cost of full inventory.

17. A computer, comprising:

one or more computer processors that execute:

accessing a demand history of inventory items;

generating, for a set of minimum (s) and maximum (S) inventory control levels (s, S), a probability distribution for a lead time demand (LTD) in future, based upon demand inter-arrival times (IA) and/or demand requisition sizes (RS) from the demand history of the inventory items, and a probability distribution for an inventory position (IP) based upon the demand history of the inventory items;

generating a probability distribution of on-hand inventory of an inventory item, for a lead time in future, directly from frequency statistics of different demand IA and demand RS on an inventory item basis based upon the probability distribution for the LTD and the IP;

computing a cost function for an expected instantaneous cost of a given IP of the inventory item, experienced for the lead time in future, based upon the probability distribution of the on-hand inventory of the inventory item; and

searching for the control levels (s, S) of the inventory item using the cost function for the expected instantaneous cost of the given IP of the inventory item.

18. The computer according to claim 17 , wherein the expected instantaneous cost of the given IP of the inventory item experienced for the lead time in future includes one or more of an instantaneous cost of outstanding backorders and an instantaneous cost of carrying inventory.

19. The computer according to claim 18 , wherein the instantaneous cost of carrying inventory includes expected cost of on-hand inventory conditioned on a current IP and/or expected full IP, and the instantaneous cost of outstanding backorders includes cost of unit backorders and requisition backorders.

20. The computer according to claim 18 , further comprising:

averaging the instantaneous costs of outstanding backorders and carrying inventory across the probability distribution for the IP; and

applying a unimodal approximation to the instantaneous costs of outstanding backorders and carrying inventory that meets one or more conditions for searching.

21. The computer according to claim 20 , wherein a total expected instantaneous cost of the inventory item experienced for the lead time in future includes a result of the averaging of the instantaneous costs of outstanding backorders and carrying inventory, and

wherein a total expected cost includes the total expected instantaneous cost and expected inventory item replenishment cost.

22. The computer according to claim 17 , wherein

the generating of the LTD and the IP probability distributions is based upon generating of histograms of the demand IA and the demand RS from the demand history of the inventory items and without assuming probability distributions, and/or by not estimating mean and/or variance parameters for the probability distributions, thereby the computing of the cost function is without theoretical probability distributions.

23. The method according to claim 17 , wherein the searching for the inventory control levels (s, S) optimizes investment across inventory items to provide lowest total backorders and/or total replenishment cost.