ZOR - Zeitschrift fiir Operations Research, Band 28, 1984, S. B297-B304.

Priority Centralization of the One Period Multi-Location Inventory System

By Abraham Mehrez, and Alan Stulman, Israel 1 )

Abstract: In this paper, we will examine a multi-center one-period inventory system. The usual penalty cost for being out of stock will be replaced by an assurance of service constraint at each location. That is, we will constrain our inventory size to meet a specific maximum probability of being out of stock at each location. The centralized system we shall propose will define a priority rule which will cause us to satisfy the entire demand of high priority locations before we begin satisfying the demands of lower priority locations. This will allow us to find a minimum initial inventory level for the centralized system that will meet all of the assurance of service constraints. We will look at the special case where the variance of the total demand of several locations is non-decreasing in locations included in the total. In this case, we will show the computations re- quired for finding the optimal centralized priority system are minimal. Finally, we will show that such a system is superior to a decentralized system.

Introduction

A recent paper by Eppen discussed the effects of centralization on the expected case of a multi-location newsboy problem. In that paper each of N individual news- boy problems was characterized by a holding cost h > 0, and an out of stock penal ty cost p > 0, which were common to all the locations. In addition at each location i, there existed a demand denoted by ~i, which was assumed to be random, possessing a distribution N (/~i' o~). The covariance and correlation between locations i and j are given by ai] and Pi/respectively, and oi//a i is assumed to be sufficiently small so that

the probabil i ty of a negative demand can be disregarded. I f H i (Yi) is the expected

holding and penal ty costs at location i whenYi items are at hand at the beginning of the period, then,

Yi oo I-1i (Yi) = h 500 (Yi - ~) ~)i (~) d~ + p f (~ - Yi) r (~) d~

i

where r ( ' ) is the density function of ~i" The results of that paper were that the

centralized inventory always has lower expected costs than the sum of the expected costs o f the non-centralized system.

(1)

1) Dr. Abraham Mehrez, and Dr. Alan Stulman, Department of Industrial Engineering, Ben-Gurion University, Israel. 0340-9422/84/08B297-304 $2.50 �9 1984 Physica- Verlag, Wiirzburg.

B 298 A. Mehrez and A. Stulman

Now it is not always possible to evaluate the penalty cost of being out of stock. In many instances, what can be provided is a demand concerning the assurance of service. We can denote the assurance of service demand by ~i, such that O~ i is defined as the allowable probability of being out of stock at location i [see Lampkin/Elowendew/ Synder].

The problem for just location i can than be defined as

Min Yi a; (Yi) = h f % - ~) r (~) d (~) (2)

Yi --oo

subject to

(~ >y;) < ~;.

The optimal solution to this problem is clearly given by

Yi = I~i + Z l - a i ' ~ (3)

and is independent of h. In a different paper [ see Mehrez/Stulman] we noted that this solution is the same as the solution to the standard formulation of the problem if a i and the penalty costs p are related by

1 - a p - - h. (4)

Thus we see that the two problems have a specific correspondence. Now if we allow the a i at the different locations to have different values, then the equivalent associated p at the various locations must also have different values. This presents a different problem from that treated in Eppen.

In this paper, we will treat the problem involving the assurance of service demands, ai, with no expression for the penalty cost of being out of stock. This is certainly appropriate in many situations (like the military), where it is difficult to assign a specific economic cost of being out of stock at any location but where being out of stock at some locations can be more critical that at other locations. We shall present a plan for centralization which if followed will lead to lower costs than decentraliza- tion. In addition we shall examine the special case where the variance of the total de- mand is a non-decreasing function of the locations included in the total. In this case, we will see that the optimal strategy is especially easy to find. For a general review of multi-echelon inventory systems, see Schwartz.

Priority Demands

A dynamic cost model with 2 items has been discussed by Kaplan. In our paper, we treat the priority structure in a different manner: A basic underlying assumption in the classical newsboy problem is that the demand and its satisfaction take place at a single instant in time, which means that the problem is only an approximation of

Priority Centralization B 299

reality. This assumption causes some difficulty in describing a centralized multi- location system. If the total demand is not greater than the total supply there is no difficulty; one just fills every order completely. However, if the total demand is greater than the total supply then it is necessary to decide which orders will not be filled. Now, in a real life situation it may be reasonable to fill the orders as they arrive until the stock depletes. This will not necessarily meet the assurance of service constraints. Also this requires us to know something about the probability distribution of the time of demand at each location as well as the distribution of the quantity of demand. This has already led us into areas foreign to the newsboy problem where, as pointed out, demand and its satisfaction take place instantly.

One way to avoid this problem is to give some priority rule to the demand from the various locations. That is, we will define rules which will say that we will begin to fill the demand of one location only after we have filled the entire demand of all the locations with higher priority. Clearly in a system of N locations there are N! such rules.

Let us now assume that we have adopted one such rule, r, and that we renumber the locations by] = 1 . . . . . n such that according to our rule location 1 has the highest priority, location 2 the next highest priority, etc.

Thus the demand from location 1 will be satisfied i fy ~> ~1. In general, the demand ,i

at location i will be satisfied i fy >~j~l ~J"

Since the random variables ~i are distributed N (/.ti, a}) then the total demand from

i locations with the highest priority will be distributed

i i i J-1 N(]~=I t.t],]~=l ~ + 2 ]=1 :~ k=l~' 0 I. O k p]k ) (5)

where P]k is the correlation coefficient relating the demand at priority location ] to

the demand at priority location k. As a shorthand notation let us define

i //(i) = ]=~1

and

o2(i)= ~ g]2+2 ~ 1-1 ]=1 j=l k=~l 01"~

(6)

Then the distribution of the total demand from the i highest priority locations is

N (p (i), a 2 (i)). The constraint that the probability of not satisfying location i be less than ai, then becomes equivalent to saying that the probability of not satisfying the total demand of the i highest priority locations be less than a i. This is the same as (2) with r (~) now being redefined asN(p (i), o 2 (/)). The equivalent solution to the solution given by (3) is

Y; (0 = ~ (i) + Z 1 - a . a (i) (7) l

B 300 A. Mehrez and A. Stulman

The no ta t lony r (/) is adopted to indicate that this starting inventory will fill the

entire demand of the higest i priority locations according to rule r, and not just the demand of the i th location as in (3).

We can now calculate y* (i) for i = 1 . . . . . N and choose

Yr = max Yr (i) (8) i

Since Yr ~ Y r (t) for all i, the set of constraints about the assurance of service at

each location will be met. Indeed we can do this for each of the N! possible priority rules and choose the rule

r and starting inventory Yr which will minimize Yr" Thus the starting inventory for a

centralized multi-location inventory system will be given by

y = rffln max Yr (i) (9) r i

The priority rule by which the orders are filled will be the rule that achieves this minimum.

Finding the Minimax Priority Rule

In order to find the solution (9) it is necessary to solve equation (7) a total of N-N! times. However, if we impose a relatively mild condition that o 2 (i) and p (/) are non-decreasing functions of i (which will be the case when means are positive and correlations are non-negative), then one of the rules can be shown to be optimal, reducing to N the amount of times (7) must be solved.

Let us assume that for some priority order r, there exists a priority position i such

that ai > 0~i+1 and thus Z 1 - ~ i < Z1 . Since by assumption p (i) and o (i) are -eti+ 1

less that/a (/+1) and o (i+1) respectively, it is clear that

* o ( i+1) Yr (i) = l~ (i) + Z 1 _a i a (i) < ~ (i+1) + Zl_ai + 1

= u 2 (i + 1) (10)

Let us examine a new priority rule, r ' , which will be the same as r except with the priority order of the i th and i + 1 st highest priority locations interchanged. It is clear that

y r , ( j ) = y ; (]) ] < i o r j > i + l (11)

Now it must be the case t h a t y r, (i) ~ Yr 0 + 1) since the a used in solving (7) is the

b * * * same in o t h y r, (i) a n d y r (i + 1) but the p (') and a ( . ) o f y r, (i) includes one loca-

Priority Centralization B 301

tion less (location i in rule r) than i n y ; (i + 1)/a (.) and ~ (.) are assumed non-decrea-

sing in the locations included. Similarly it must be the case thaty r , (i + 1)~<y; (i + 1). Since in solving (7) they

both include the very same locations but the Z 1.~ corresponding to y; , (i + 1) is less

than the a corresponding t o y ; (i + 1).

Combining all these results comparing r' to r, we see that it must be the case that

Max yr, q)<~ Max Yr q) (12) I ]

Thus, the priority rule r' is preferable to the priority rule r. We can continue this procedure repeatedly until we will arrive at the minimax

priority rule r*. This is easily seen to be to order the priorities inversely to the ~i" That is, the location with the highest priority corresponds to the location with the smallest ~/, the location with second priority corresponds to the location with the next smallest a i, etc.

Thus, if we are willing to make the mild assumption that/1 ( ' ) and o ( ' ) are non- decreasing in locations included, it is sufficient to calculate (8) only for the order r described above. This requires (7) to be solved N times instead of N-N! times.

To Centralize the System or Not

We have seen that (9) is the required starting inventory in our centralized priority system, and that all of the assurance of service demands will be met. We still need to examine whether it is more economical to have a decentralized inventory or to centra- lize the inventory according to a priority rule.

It is clear that two systems that meet all constraints need only be compared with respect to the expected number of surplus items left after the period. Now regardless of which system (decentralized or priority centralized) is chosen the total demand is unaffected by the choice. In a centralized system the surplus will be the larger of zero or the starting inventory minus the total demand. In a decentralized system the sur- plus may be larger than the greater of zero and the starting inventory minus the total demand. This is because at one location there may be excess demand while at another location there is a surplus. Thus, it is clear that if the centralized priority system required a smaller starting inventory than the decentralized system it would necessarily be the more economical.

From (7) we have that the total inventory required by a decentralized system that satisfies constraints is given by

N N Yd=i~= l lli + i~= l ZI_~. o i (13)

The starting inventory required by a priority centralized system is y* given by (9). We will now show that it is always true thaty d >y* .

B 302 A. Mehrez and A. Stulman

Consider the priority rule discussed in the previous section where the priorities are defined according to increasing values of % Only if the special conditions of the last section are met will this be the optimal priority rule. However, regardless of whether this rule is optimal or not, let us define y : to be startinginventory required to satisfy the constraints if this rule is used for setting priorities. Ya is calculated as in (8). That is,

Ya = max Ya (i). (14) i

Let us denote the locationj such thaty a =Ya (])" Then we have

N N Yd =is 1 l~i + iZ= 1 Z l - a i o i

~ l a q ) + ~ Zl_ot i i=1 o i +

i = 1 ZI-~ i o i

N ZI_ot i ~

i = / + 1

(15)

But by the priority rule being used, Z 1 -~k for all k <]'. Then continuing >~Zl_~ from (15) we have,

J Yd > g q ) + Z x - ~ / i ~ l ~

(16)

Now the variance of the total demand of the first j priority locations is given by

/ o0r 1 ~ + 2 ~ ~ _- .< J oj Pii ( iv)

i ~ k /

and the square of i ~1 ai in (16) is

/ J = o 2. + 2 "s Z o i (18) ( i ~ 1 ~ i~=l z i4=k ~"

i ,k<~j

Since Of < 1 we have

/ J o (])2 < ( t '= l ]~ ~ = ) ~7 q) < i = 1 ]~ fli (19)

From (19) and (16) we have,

y a > u ( / ) + z a _ 5. ~

Priority Centralization B 303

or > *

Yd Ya (20)

Now Ya >~Y*" Thus we have the Yd >Y*" In words, the initial starting inventory

required by a decentralized system that will satisfy constraints is greater than the required starting inventory of the optimal priority centralized system that will satisfy those same constraints. Thus the optimal priority centralized system is the more economical of the two.

Concluding Remarks

In this paper, we have examined the multi-location, one period, inventory system without out of stock penalty cost, but with assurance of service constraints. We have seen that there always exists a centralized priority system which will meet all of the assurance of service constraints and which will be superior to a decentralized system that will meet those same constraints. We also saw that under some mild conditions, it is a simple task to determine the optimal system.

Note that this result is broader than the result in Eppen. If all the locations have the same constraint our problem appears to be the same as that treated by Eppen. However, we allow the constraint to be different at every location. This means that the superiority of a centralized (priority) system has been shown for a broader class of situation. We have not examined the situation where there are different penalty costs at each location instead of assurance of service constraints. However, it would be a pretty good guess that some kind of centralized priority system will again prove optimal.

Another approach of the problem of centralizing a multi-location one period inventory system would be to fill orders according to some random mechanism when the total demand exceeds the starting inventory. It seems intuitive that here too it should be possible to meet the assurance of service constraints with a lower initial inventory than that required by a decentralized system meeting those same constraints. This system still requires study.

Still another area of study would require observing a centralized and decentralized system with the time element included. That is where a period is some fixed time r longer than an instant. In such systems, we would have to take the time of each de- mand at each location into account. However, this would allow us to explore centra- lized systems where demand is Filled on a first come, first serve basis.

A Short Numerical Example

In this example, let us assume four locations with independent demands. The follo- wing table contains the input data for this example.

B 304 A. Mehrez and A. Stulman

Location

1 2 3 4

a 0.01 0.05 0.05 0.1

# 50 60 60 50

a 10 10 10 10

Since the location demands are independent of one another, it is clear that o (i) is a non-decreasing function of the number of locations included. Thus the optimal priority rule is to order the priorities according to increasing ai" That is, location 1 is given the highest priority, location 2 the next hi_gl~,est, etc. �9

It is now necessary to calculate the values o f Z i for i = 1 . . . . . 4. We getZ1 = 73,

Z2 = 133, Z3 = 198, Z4 = 237. The priority centralized starting inventory is then Z* = 237. The total starting inventory required for a decentralized system that will meet all the constraints is 290.

Eppen, G.D.: Effects of Centralization on Expected Costs in a Multi-Location Newsboy Problem. Management Science, 25, 1979, 498-501.

Lampkin, Ir and Elowendew: Computation of Optimal Re-Order Levels and Quantities for a Re-Order Level Stock Control System. Operational Research Quarterly, 14, 1963, 263-278.

Mehrez, A., and A. Stulman: Another Note on Centralizing a Multi-Location One Period Inventory System. Unpublished Working Paper, Dept. of Industrial Engineering, Ben-Gurion University.

Synder, R.D. : The Safety Stock Syndrome. Journal of Operations Research Society, 31, 1980, 833-837.

Kaplan, A. : Stock Rationing. Management Science, Vol. 15, 1969, 260-267. Schwartz, L. : Multi-Level Production/Inventory Control Systems: Theory and Practice. Tiros

Studies in Management Science, Vol. 16, 1981, North Holland, Amsterdam.

Received on April 26, 1984.