1. OPTIMAL POLICIES FOR A MULTI- ECHELON INVENTORY PROBLEM ANDREW J. CLARK AND HERBERT SCARF October 1959 Presented By İsmail Koca

2. What is the Paper About? Determining optimal purchasing quantities separately for all levels of installations in a multi-installation model where the model consists of several installations, say 1,2,...,N with installation 1 receiving stock from 2, with 2 receiving stock from 3, etc.

3. NN-121 Model Type 1 Model Type 2

4. DIFFERENCE FROM OTHER WORKS Previous works: determine the optimal purchasing quantities at a single installation This work: considers multi-installation models’ optimal purchasing quantities where lead time is also affected by the availability of stock at the supplier installation.

5. APPROACH FOR SOLUTION 1.Define a cost function for each configuration of stock at the various installations, and in transit from one installation to another 2.From defined cost function do a recursive computation for the optimal provisioning policy.

6. ...  in practice this computation of a sequence of functions of at least N variables is difficult and impractical  this work simplifies this computation without compromising the optimality of the solution by several very plausible assumptions

7. ASSUMPTIONS Demand originates in the system at the lowest installation Purchasing, shipment costs are linear. No setup cost except the last installation

8. ... Holding and shortage costs are assumed to be function of echelon stock which is the stock at that level plus all all other stock in the system which is actually at a lower level or in transit to lower level. Each echelon backlogs excess demand

9. COST FUNCTION c(z): cost of purchasing an amount of z : lead time  (t):density function of demand (may differ from period to period) (t: demand) h: holding cost per unit p:shortage cost per unit x:stock on hand at the beginning of the period

10. Cost during the period (exclusive purchase cost): hx + p x>0 L(x) = p x  0

11. Discounted cost function for n periods x 1 : stock on hand w j :units to be delivered j periods in the future  : discount factor

12. In the above function it is assumed that all excess demand is backlogged until the necessary stock becomes available Minimizing value of z is the optimal purchase quantity for the given stock configuration

13. An open form of the above formula is:

14. and f n is:

15. order policy... The discounted cost is assumed to be convex – True if holding and shortage costs are linear Then there exists a sequence of critical numbers (S n, s n ) Order if x 1 +.....+w -1 < s n and with an amount of S n -(x 1 +.....+w -1 )

16. Example: Results are shown by means of an example  2 installations  Lead time: 2 periods  x 1 :stock on hand at installation 1  w 1 :stock to be delivered one period in the future  x 2 :echelon 2 stock  L(x 1 ): one period costs at installation 1  (x 2 ): one period costs at echelon 2

17. Optimal policy for the lowest installation For n>2 the discounted cost for the example is: and f n (u) satisfy:

18. if x 1 + w 1 < minimizing value is if x 1 + w 1 < ordering occurs and the minimum cost will be:

19. if x 2 < we are only be to ship x 2 – (x 1 + w 1 ) and therefore the minimum cost will be:

20. ... This cost is larger than the previous one. The insufficiency of stock level at 2 caused additional cost, which is one period loss to be charged to this echelon. The difference is: if and zero if

21. ... By adding this one period loss to the second echelon, the optimal policy is then computed using the formula given. Also as the previous function is a convex function of x 2 the optimal policy for second echelon is will be of type (S,s)

22. Optimality of the formula for model type 1: There is a sequence of functions g n (x 2 ), with g 1 (x 2 ) =, such that NN-121

23. ... where  n (x 2 ) is: for and zero for is a function of x 2 alone

24. and… The solution of this equation provides us with the optimal policy for the entire system

25. Optimality of the formula for model type 2: A1A1 C1C1 B1B1 B2B2 A3A3 A2A2

26. The assumptions for the previous model is retained for this model There are 2 points: 1.It is not permitted to exchange stock between any arbitrary pair of installations as it does not run contrary to what is in practice 2.If all requests cannot be satisfied because of insufficient stock at a higher echelon, how is the available stock to be rationed among the requesting installations?

27. After some iterations similar to the first model we see that C n (x 1,x 2,x 3 ) cannot be broken down in the form of

28. If this was the case at the end of the solution of the minimization problem than there is so far so good solution. The problem arises when the equation above is as:

29. … This is the case pointed out in Point 2 above. In order to allocate the insufficient stock between the sub installations we should solve the minimization problem for the cost function. The solution of the minimization problem is good when we lower installations are not out of balance. And this is expected to occur rather frequently

30. So the approximation to is an excellent for this model..

31. EXTENSIONS If the demand at echelons are dependent to each other we use joint density function of them in formulas instead of  (t). This does not change the general structure of the formulas, so the minimization for the new function works similar to the model 1’s solution. Then gives good solution.