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^¦¦^.¦^r^. ..¦..,¦ .:tv.. mmaammmm ^an¦ —¦
SyfJ9 .P^
AN OPTIMAL INV^'TQRY POLICY FOR A
MILITARY ORGANIZATION
Edward B. Berman
Andrew J, Clark.
P647
March .30, 1955
.. , As
¦ .
?<&mm(?ÖX/t01> ^. ..^^.^
P647
for each spare part which could cause the end item to be out of commission
for lack of the part, is, in a sense, the contribution of the spare part to
the pool of end items out of commission« Thua. no matter how many of the
spare parts are stocked, there is some probability of these not being enough
to keep the end item in commission, and therefore, each spare part has some
expected contribution to the,pool of end items out of commission for parts.
.¦...¦^.1[ MM—1
P647
6
PAHT I  THE BASE
2* THE GENEEAL SOLUTION FOR THE BASE1
In this section we consider the base as separate from the system» Implicit
in the derivation of the base policy, are the following assumptions,
most of which have been already suggested in the introduction:
A, The total issues which occur during the day are assumed to have
occurred one at a time, and evenly spaced throughout the day,
B, The balance on hand is compared with the reorder level after each
issue, and a requisition initiated if the balance on hand is equal to the reorder
level. The requisition, each time, is for an amount, A , less any
amounts received on priority requisitions during the previous process of replenishment.
Also, the stock on hand when the order arrives is not less than
the reorder level.
Assumption A and B taken together permit the further assumption that
the balance on hand is exactly equal to the reorder level at the time the
requisition is submitted,
C, The ability of depots to fulfill requisitions is assumed infinite.
This does not imply that such is the case, but rather that any costs arising
from depot failure should be assessed against the inventory policy of the
system as a whole.
Assumption C allows us to look at the base as an isolated supply
The general approach taken in the base solution is similar to that of
T, M, Whitin, Theory of Inventory Management,, pp 566?, We are indebted to
T, M, Whitin for a critical review of a preliminary draft of this part of
the paper. We are also indebted to R, Bellman, 0, Morgenstern, and others
for their comments on the preliminary draft of this paper. Perhaps we
should also express our appreciation to K. J. Arrow, T, Harris, and J.
Marschak, for it was their paper. Optimal Inventory Policy, Econometrica,
Vol. 19, 1951, July, pp .?50?7?, which initially introduced this problem to us.
,.,. M—MMMWfl
P647
7
activity for our initial derivation. This assumption is modified when we
consider the base as a component of the system,
Ds Whenever demand exceeds the balance on hand, an emergency requisition
is submitted for the excess of amounts demanded over amounts on hand. The
emergency requisition requires premium communication and transportation.
E. The routine pipeline time, p, and the emergency pipeline timö, p,
are assumed to be constants.
Next, we will define the various costs encountered in the operation of
the base, together with the symbols to be used in the derivation of the base
policy« First, however, we should clarify our position in one respect. We
derive our stuckage policy on an individual item of supply basis; hence, our
various costs are also on that basis. Since many of the costs are reduced,
per unit, due to aggregation of items (such as the fixed paperwork cost of
requisitioning, wherein if several items are included in the requisition, the
per unit cost is less than the cost of a single item on the requisition), we
assume that such reduced per unit costs areised. In practicej these reduced
costs due to aggregation can be found by sampling techniques.
The symbols and component costs of supply used in subsequent calculations
are:
(l) k ¦ fixed costs of handling the paperwork and communications for a
routine requisition.
(?) & = fixed amount of reorder.
(3) p, + p?£i = packaging, inspection, and handling costs, where p is the
fixed component and p is the per unit cost associated with
the size of the order, ^.
(4) t, . t.a =• cost of transporting an amount ^ from the supply depot to
the base.
 • ¦ MIW—¦
s P~647
8
(5) 7 ^ amount of stock on hand.
(6) d^ + 6.27 = warehousing, depreciation, and obsolescence costs per unit
of time for holding an amount ye These costs are also
referred to as "holding" costs«
(7) 0 " amount of depletion.
(8) p = routine pipeline time (time from when the balance on hand reaches
the reorder level to the time of receipt of the materiel) ¦¦ constant.
(9) p ^ priority pipeline time (time from the indication of need for an item
unavailable at the base to the time of receipt of the item) ¦ constant.
(10) b0 + b]_q + b2pq "¦ depletion penalty, or the cost of understocking the item.
In this cost, b0 is the fixed cost, if any, which is
independent of the amount of depletion. The cost bn
is the cost associated with the amount of depletion
but not length of depletion. The component b^ p q is
the cost incurred through loss of utility of an end
item being out of coramissioh for p days. Here b
could be the cost per day of having an extra end item
available for use. The component costs of the depletion
penalty will be discussed in further detail later.
(11) x IS number of the item demanded per unit of time. In subsequent work,
x is a random variable.
(1?) f(x)  demand probability (density) function of the random variable, x.
This function may be either continuous or be defined for only
integral values of x.
(13) F(x) ^ J f(t)dt a the integral 3r cumulative demand probability
0
function. Since f(x) may be either a continuous'
function or a density function, such as a Poisson
distribution, f(x) must be integrable in the
..—UP, —.¦.,.*.—1^
P647
9
Stieltjes sense over all intervals on the
positive xaxis with zero as a lower limit. We
also have the restriction that F(x) —5> 1 as
x —^> oo.
(14) R = reorder level. This is the point in the stock inventory where a
routine requisition is required.
(15) 6 a time period between the arrival of two successive orders.
We notice that most of the costs encountered by the base (in particular,
items (3), ih), (6), and (IO)above) are expressed as linear, nonhomogeneous
functions of the form y » a + bx. These costs, in reality, do not assume
this form, but may be reasonably approximated by such functions. Actually,
in our later formulation, we could just as easily consider these costs to be
expressed as arbitrary functions expanded in power series. However, in
practice, probably the best we can do is to find the linear approximations
to the cost functions; hence, we restrict ourselves to such linear cost
functions in our formulation.
In order to obtain a clearer understanding of the situation and to
facilitate the calculation problem, we shall take the routine pipeline time
as the fundamental time unit. This practice, in fact, is one of the main
characteristics of our approach and provides a significant simplification of
the problem. This unit, then, will also serve in the definition of those
parameters defined in terms of units of time, such as the parameters d, and
d , the variable x, and the requisitioning period, 0, Thus, for example,
we speak of Ö as being "so many routine pipeline times".
Our base inventory as a function of time may now be portrayed graphically
as shown in Figure (1). Figure (1) represents a case of reality, where issues
can be made several at a time and at any time during the day. By applying
P~647
10
assumption A, we amend Figure (l) to appear as shown in Figure (2), Thus,
the issue of three items, causing a drop in the inventory from A to B in
Figure (1), is represented in Figure (2) as three separate issues of one
item each and at equal intervals throughout the day. Also, note that in
Figure (2), the routine pipeline time, p , is considered to tie an integral
number of days; this is not at all necessary, but in practice such would
probably be the case«
We are now in a position to write a function representing the cost of
operation at the base for a given item and for a typical requisitioning
period, 0:
(1) L(0,R,A) = (dx + day) 0 + k M'Px + ?2^ + (tl + 4^ + bo f1 ~ F (R)"
»oo
+ (b, + b2p) J (x  R) dF(x).
R
The first term of the right member represents the cost of holding a
quantity y in stock over a period of time 0. The next three terms are the
handling, packaging, and transportation costs for the routine requisition. The
term b fl  F(Rj represents the expected value of those costs of depletion
which do not depend on the length of depletion nor amount of depletion. In
this term, b is the fixed cost of depletion and 1 1  F(R)1 is the probability
of incurring the depletion, since F(R) represents the probability of issuing
R items or less during the pipeline period when routine replenishment is occurring,
and I 1  F(R)"I represents the probability of demands being greater than R during
J /TOO
that period. The term (bi + bp) J (x  R) dF(x) represents the expected costs
^ R
of depletion which vary either with the quantity alone or with the quantity,
and duration of depletion. In this term, bn represents the costs which vary
only with the quantity of depletion; b9 p represents the costs which vary with
both the quantity and the duration of depletion multiplied by the length of
""—*
•r
¦j ' :
s ¦.. .
:;•'
: }
t
'A'
' 5
' 0 
ill
;
¦ i
P647
12
depletion; (x  R) is the quantity of depletion for a demand x; dF(x) is the
probability of the demand x ; and the integral from R to oo represents the
summation of the products of all possible depletion sizes and their probabilities.
Note that the two terms representing depletion costs are valid only
under assumption B, in which it is assumed that the requisition is initiated
when the balance on hand is exactly equal to the reorder level, R.
Now we shall consider a number of consecutive requisitioning periods,
ÖJ! (i = 1 to n). But first, in Figure (2), let us join with a straight line
the stock level C at the beginning of the requisitioning period and 'the
stock level D at the end of the requisitioning period. This line might be
considered, as representing a kind of "average" demand during the period 0,
If we construct such lines for the consecutive requisitioning periods
0. (i ^ 1 to n), we obtain the following picture:
Figure (3)
R + A
Amount
on
Hand
routine pipeline time
From Figure (3)j we can get a clearer picture of our problem. If R is
high, then we incur more holding costs. If R is low, the possibility of incurring
the depletion cost is increased. If A is large, our holding costs
¦a—i——B>B—a
P64?
 13
are again increased. If A is low, we requisition more often and increase
the costs of requisitioning. Our problem, then, is to find the value of R
and A which minimize the expected cost of supply per unit of time.1/ To find
the expected cost of supply per unit time we must average the cost of supply
over many requisitioning periods to allow for the variance of one requisitioning
period from another. For this reason, we cannot merely minimize the cost
expressed by equation (1), since this is only the cost of one requisitioning
period.
Let T represent a period of operation of the supply activity, so that
T = 6u + 0 + ... + 0 ¦ E 0., where 0. is the time period between the
12 n i:al i'
ith and the i+l~th receipt of materiel. Then the total cost over T is
n
given by E L(0.,R,A), where L(0J,R,A) is the cost over the time period
il
0^ as given by equation (l). The cost per unit of time is then:
n v n L(0,,R,A)
cost per unit time " X ^ = n
4% § ^ x n
.Eei
where we have divided numerator and denominator by n and let 0 » lkü± >
n
which is the average requisition period. Substituting from equation (l), we
now obtain:
(2) X  (d1 + d2y) + 1 jk + (p1 + ppA) . (^ + t^A) + b0 Tl  F(R)1
1/ We might remark here that we are interested in minimizing the averaged expected
cost per unit of time rather than per requisitioning period. It can
be shown that the value of A which yields least cost per requisitioning
period is zero. This implies the average requisitioning period is also zero
which in turn implies all requisitioning periods are zero. Thus, we do not
obtain a useful solution to our problem.
aaaaa a
—¦¦" ''' '"• '••¦ nmm M MM in
P647
14
9
where we have assumed y and A to be constant relative to the summation.
«
In equation (2), y becomes the average amount of stock on hand per
unit time, obtained by including experience throughout T, The next thing
we must do is express y in terms of R, A, and known quantities. To do
this, we introduce the notion of an average requisitioning period, The
length of this period is just 0, of course. The stock on hand at the beginning
of this average period is R +A  x, where x is the average of the
demands during the n routine pipeline time intervals, which immediately
precede the receipt of the materiel; »Similarly, the stock on hand at the
end of the average period is R ~ x.
Now let the number of requisitioning periods become infinite. Then
oo
x ~^£ = f x dF(x), which is the average expected demand per routine
o
pipeline time. Also, the balance on hand during the average requisitioning
period will appear as follows:
r^i
•*
Figure (4)
c j
z
k
Amount
R+A6
\
on
Hand •. 1
^ 1
R
1
o Ü
Tljns
N'otice that the stepfunction character of Figure (4) is due to the fact that
P647
15
we issue one item at a time, and cannot issue fractions of items. If the
amount on hand were a linear function, as represented by the diagonal dotted
line in Figure (k), the average amount on hand would just be R  6 + ^.
However, we must add a correction term of « to allow for the stepfunction
effect. Thus, we obtain
(3) y  R  £ + Ä+1 .
The value for y in equation (3) validates our prior assumption that
j be constant relative to the summation in equation (?). It might also be
noted that we are charging the warehousing costs for only an amount y whereas
warehouse space is needed for the maximum amount stocked. We feel our assumption
justified on the basis that all items in a warehouse will not be stocked
in their maximum amounts at the same time, but will indeed average out as
assumed, with some items requiring space for more than the amount assumed and
others less in any given period of time.
Next we will establish a relationship between A and Ö. If we divide T
into n equal intervals, each of length 0, we can write
5o
Z x..
. T ik
x. = k1
i
9
as the arithmetic mean demand for the ith period. Therefore,
0
0 Xj ~ , L, X.. and summing over i, we get
1/ A more rigorous development of equation (3) is as follows: Define the requisition
period, 0., to be from the ith time the balance reaches the reorder
level to the i+lth time. Then the average stock on hand during the kth day
of 0^ is given by
k
Hk ' R  ^ xii + ^ik + i (i < k < P)
where we are temporarily letting ;< represent a dayts demand rather than the
n n 0
9 £ x « £ ^ X4i
n 0JL
E E x = n A,
which is the total demand during T. Employing the extreme left and right
members,
(4) A  0 i«l
n _
E x.
i 0 x.
n
Estimating x by £ , we substitute the values for y and 0 from
equations (3) and (4), respectively, into equation (?) and get
f{k + p + t, + bn)
X(A,R) = d1 * d2(R  e + äl 1) + 1 +£(p?
+ tp)
2 A
^ b0 ^ (b, + b? p) ^ Qo
_F(R)+ J' (xR)dF(x).
A A R
P647
16
demand per pipeline time as previously defined. To show the derivation of
these equations more clearly, we might isolate the kth day, which would
appear as follows:
yik'
k
R  £x. . + lx.w
j we infer that
the balance on hand goes negative in response to demands after total stock
depletion. In actuality, balances do not go negative; but the correction term
to be added to the value for y which we obtain by allowing negative balances,
can be shown to be:
.. t. in HI .' i^jv:,.;¦;.¦,—i^i. , ¦ r;.. ¦¦¦ ' — ^  " ¦¦"i:—"' ¦ ' " ' —.. .„>..^'.
P647
QP 18
d? f(B + bn)  £b0F(R) + izS (x  R) dF(x)
(7) ex « _  „0 ^ R  o
We may solve explicitly for A in equations (6) and (7) and get
Q0>
(Ö) A  i_Un f(R)  C A (x  R) dF(x)
d2l Ü OR JR ^
2 ^ ,. _ o00 ¦>
(9) A = 26 ) B + b0 [l  F(R)J + C j (x  R) dF(x) .
drj ( R J
Now if A is eliminated from equations (8) and (9), the optimal reorder
level, R*, may be determined from the equation which results and substituted
back in equation (8) or (9) to find A'f , the optimal amount of reorder.
Whether or not the values so obtained yield the actual minimum supply cost
depends upon an investigation of the second partial derivates. We will omit
this investigation: for all reasonable demand functions, a valid minimum does
exist.
L ) (x  R) (x  R  1) dF(x).
A " R 4x
This term, upon inspection, is so small that it can be comfortably neglected.
Returning to equation (3b), we take the weighted average over n requisitioning
periods and obtain , < if " ' ¦~,~^'r'~f~io
S 0 y. n ^i (j  1) x ^piS^'J^vWJ
(3c) y ¦ i°l i .1  R + 1  A + ^ 7 
% ~ 0 ' npO
By resubdividing our interval T into n equal intervals, each of length 6,
we can rearrange terms in the last member of the above equation to get
n x..
(k  1) Z _
2 i1 n
n i1 jl
n &i (j 1) x.. n p9
E E 2 iJ  E 2
(k  i)xik pe
 E
¦1 .i1 nnft ivl lc"1 npS k1
p9 (k
= E 2 K
k1
pO
pO
But x, ^ I for each k as n ^ oo. Substituting, we get
taa i—aaao——wwi
P647
19
The calculation of R* and A^ from equations (8) and (9) is a
relatively easy matter, particularly on the electronic calculators. In fact,
for some simple demand functions, explicit answers for R* and A* can be
obtained. Examples of such functions are discussed in a later section.
The case of recoverabletype items at the base can be included in the
above results by just considering the demand function, f(x), to represent
the net loss due to items condemned and items beyond base repair. This is
due to the fact that if an item is basereparable, it can readily be converted
to a serviceable item  more readily, in fact, than obtaining the
item from any other source. Hence, a basereparable item can be treated as
s erviceable from a stock policy point of view.
In the derivation of our base results as expressed by equations (8) and
(9), we used the idea of the number of requisitioning periods becoming infinite.
One might question whether the results are tenable based upon such processes,
inasmuch as it is certain the supply system will not operate forever. Actually,
n pei (J  IK. pö (k  1)*
i^l .I"!
1,1
npÖ
k"l
0
[pQ (pQ^ 1} pQ]6
& Pp^Q
Substituting back into equation (3c), we get
(3d} 7  R + 1  A. * £ Q .
2 0
Subsequently, we show that A « £ 0 so that equation (3d) becomes
y  R  i? + A+l .
...„..—¦ 1
P647
20
however, the method is validated by an implicit assumption, namely that" our
demand probability function is quite independent of time. Thus, even though
reason might assess a zero probability of issuing n items a« thousand years
from now, this probability is not reflected in our demand function. Indeed,
our demand function assesses the probability of demand to be the same for •
all time. But since our problem was to establish some A and R to use in
the stockage policy, the use of A* and R* as calculated abovp is the best
we can do if the given demand function, f(x), is the best estimate of demand
probability that we can obtain. If the demand probability function can be
expressed in terms of time, then this becomes another problem for which we
have no general solution. On the other hand, we can approximate a. solution
by recalculating R* and A* periodically with a demand function adjusted
to reflect the trend established by past consumption and other factors. The
derivation of the demand probability functions, however, is beyond the scope
of this paper.
3. ROUNDING TO INTEGRAL VALUES
For practical application of the optimal reorder level, Rw , and reorder
amount. A* , we must examine the problem of rounding them to Integral values.
If we construct our cost surface, X^R), in the vicinity of (A*,R*). we might
obtain the situation shown in Figure (5)»
Figure (5)
A(A*,R*)
By^ 7
AfM AM
A' A* A»+l M^ ^
/ /
P647
/ ' 21
In Figure (5), Rf and A' represent the largest integers contained in
R^ and A^ respectively. The values, RT, R» + 1, A«, A» + 1, determine
the rectangle ABCD on the cost surface. At first glance, it might seem
justified to round A# and R* to those integral values o.f A and R
which yield the least cost among the four values, ^R1, A')» ^RSA1»!),
}v(R»+l,A«), and Xß^ljA'+l) which occur at the points A, B, C, and D. On
the other hand, it is entirely conceivable that the cost surface, X(A,R),
in the vicinity of (AW,RW) might appear as follows:
Figure (6)
X(A*,R*)
equal cost contour
From Figure (6), it is clear that it is quite possible for the least cost X,
for integral values of A and R, to occur not at the corners of the
rectangle ABCD containing }[ A*,R*), but at the points E or F or, in some
cases, at even more remote points on the surface»
In general, then to find the integral values for A "and R which afford
the least cost of supply, wo must proceed as follows: First we find that
value, A. , which minimizes the value >iR * R'^). We do this by substituting
R* for R in equation (7)j the partial derivative of X with respect to A,
and solve for A , the solution being A., . We then repeat this process by
substituting R» + 1 for R in equation (?) to find A;>> that value of A which
 "•' "*"
P647
~?2~
minimizes X(R ^ R?+1,A). Continuing this procedure, we substitute A1 for
A in equation (8), the partial derivative of X with respect to R, and
solve for R^, the value of R which minimizes X(R, A0 Af). Similarly,
we find R2, the value of R which minimizes X{R, A ^ AT + 1). We have
now found the points (R», A.), (Rr + 1, A ), (R1, A»), and (R , A? + 1) on
the X  surface, which represent the minimum values for X along the extended
"sides" of the rectangle ABCD in Figure (5). If we let the symbol
[xj represent the largest integer contained in x, we can now set down the
following eight values for X ;
(c^ XR», CAJ ) (c5) x( [RJ, A»)
(C?) XR», [AI>1) (C6) XCRI>1I Af)
(c.3) X{R^I, [A^}' (C7) XCHJ* Af+i)
(c4) XRHI, [A?>I) (C8) CVKA^D
Of the eight values of X so obtained, the least one provides the integral
values of A and R to be used. The conditions on the nature of the surface,
X(A,R), which validate this process are assumed to exist and do exist for
any practical case.
It might be remarked that the situation depicted in Figure (6) can occur
in practice only for low values of Aw and Ri:'. For large values of A*
and R^, the cost surface becomes so flat that one can automatically round
to the nearest integer. For a region of the surface between these extremes,
only the four values, X(Rr,At), ^R'+l^A1), A(Rf,A,+l), and x(R»+l,A»+l)
need be computed and compared.
23
4. DETERMINING WHEN TO STOCK AT THE BASE
Since our cost .function, >(A»R) in equation (5), expresses the cost
of holding an item of supply at the base, it is invalid for A ^ 0S R = 0,
This situation would be interpreted as not stocking the item at the base at
all, but stocking the item at the depot instead. The average cost of supply
per unit time, in this case, becomes
(10) /\  £(br + b. + b p)
0 1 '; which is nothing more than the depletion penalty multiplied by the average
probability of incurring it per unit of time. A superposition of the two
cost surfaces represented by equations (5) and (lO) might appear as follows
Figure (7)
In Figure (7), the hatched area of the AR plane represents those values
of A and R for which the X  surface lies below the t)  plane . From
 . . ^ P647
24
equation (5), we also notice the cost X approaches infinity as Ä approaches
zero, which renders the X surface invalid at (0,0). Of course this situation
is reasonable; if A approaches zero so does 0 and the fixed costs for each
0j occur more and more often until in the limit they occur infinitely often.
In Figure (7), it is clear that, with the scale values chosen, the cost ¦
A on the X  surface (which is yielded by A ~ 1, R a 0) would be chosen
for any optimal A* and Rw in the crosshatched area« A very simple method,
however, can be established to decide whether to stock at the base or not.
This method is to compute the optimal, rounded Att and R» as outlined in
sections 2 and 3S and then compare the resulting cost X with the cost, /]_ 5 .
of not stocking at the base. We notice, in this connection, that the cost f\
is easy to calculate, being nothing more than the cost of depletion multiplied
by C .
From our expression for X in equation (5), it is fairly clear that the
possibility of not stocking at the base can occur only for highcost, low demand
items or for low demand items with low depletion penalties. For lowcost,
low demand items, the constant costs of requisitioning, in addition to any
relatively high depletion costs, will cause the items to be stocked at the
base. In fact, a high depletion cost for lowcost items will cause large
stocks at the base even for very low demand rates.
5. OPTIMAL POLICY FOR UMKNOWft MEAN DEMAND
If, in the results of section 7!, the form of the density function f(x)
is known but the mean demand € can be expressed only as a probability
function, g(t), we can extend our theory as follows: First we calculate our
optimal cost as a function of C :
(11) X(6) ¦ >(AK,US6).
Then, for each optimal &« and H we multiply the cost \(0 by the prob
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25
ability of incurring that cost and sura over all such probabilities. In this
manner we obtain
(12) r(f)  J g(t) ^A^),RK6),t) dti
in which we changed the variable 6 to t for the integration and remembered
that A* and R# are themselves functions of 6, That value for 6, then,
which gives least cost is calculated by the usual procedure of setting the
first derivative equal to zero* Therefore, optimal A* and Rw are given
by
tft m A«(6*)
(13)
R*  R*(£*),
where 6* is a solution of the equation
oo
dYtO * 0 »± f g(t) ^A^)^^)^) dte
d € d 1, it is clear that minimum cost can occur only for R ¦ 0, or
Pö47
28
R ~ 1, if only integral values for R are allowed. By direct substitution
in equation (5), we get
d,
1 A
X(A,R«0) = A + _ A + ß(T tMl
(15)
X(^R«1) « A + d ( A + 1) + ^T
? A
where T = k + ?,_ + t^ and TT = bo + C  b0 + b1 + b2p » total depletion cost.
Equation (9) yields the values
A1"" «,
0 \
2J (T ¦)• ßTTJ (using R « 0)
d2
A«
1
\
2J3 T (using R = l)
d2
It is clear that the equation X(A* R««0)  X(Äf,R*l) yields a ß^ such
that for ß < ß?, R » 0 with A^ provide least cost, and for ß > ß», R = 1
v/ith A^. provide least cost. A calculation shows, in fact, that ß* raust
satisfy the equation
DVT? dj 2 = 8 d2ß»T.
For rounding, we compare the four values
^•(0^1 Ro)
X([A^I, R=0)
XCAJG, Rl)
XC^I, ^1)
and choose the integral A* and R" which provide the least value. The least
.value sn obtained, X(A:;,R8) car. then be compared against the cost f] •
ß(b + b, + bpp) to decide whether or not to stock at the base. We would not
—1.r..
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29
stock if \ < X(A%R"0e
Let us now consider the case where ß , the arithmetic mean of f(x), is
given hy a probability function, g(t). Equation (1?), together with
equations (15) give
riß)  A+ ^A*(ß) + KT^/T (RS=0)
2 [' AgTFT K J
Y(ß)  A + d2(
AJ(ß)
2 ÄKßT
where ^oo
i »y t g(t)dt
DO
t g(t)dt.
The derivatives with respect to ß yield
x)dx to be 1.
a
r
Substituting f(x) • o^~ax and F(x) • I f(t)dt ¦ 1 
"'o
equations (8) and (9), ?nd eliminating A , we obtain
2d.
ax . ,
e into
r aß»
(bn0 + C ) V a a
aRtf /(b0 + C) + B "j .
Let z e"^ (b +C). Then
0 ä
z 
2d.
a
2d
z  _ B
a
0,
and r z l+/l + 2aB r 1/ i
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31
Since z is positive and all terms under the radical are positive, we take
the plus sign for the radical and get
OR» . V + C
d r 1 + , /l + 2aB 1
having replaced our value for z. When substituted back into equation (8),
we get
e"0^ (b (i + C) L *,/! ^ /aß
A^ « 0 / d_2 »
od. a
Also, the optimal average requisitioning period is given by
Q% => Af => aA* « 1 +i/ 1 + 2a B.
e * d0
We can summarize our results by the equations
+ V1+ r
A« = QK
a
P.* « 1 In
a
b a + G
0
d 0W
Integral values for Aw and R» can now be found by the method of
section ;., For the situation where the mean, 1, can be expressed only by
a
a probability function, git), an explicit answer for optimal A and R
cannot be obtained; in practice, a numerical method would be used as suggested
in section 5»
ri ¦ ¦ iniii —Ml
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32
Sxample 3> Let the demand probability function be of the form
(20) f(x) e" m (Poisson distribution).
xl
The Poisson function is useful to express most demands at bases, provided
it is used on an item of supply basis as is done in this paper« This'function,
in particular, is much more realistic than either of the other two examples
for higher demand rates*
oo
We see that this function satisfies our condition that j f(x) dx » 1
..oo
and we also calculate (R,A) • d_  M + d. (R 6 . A . 1) + * C(p * t )
^ ^ 2 A ' '
+ ^P^  _ FCR) ^ !£ J (X  R) dF(x),
A A R
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37 v
having substituted A for 9. Thus, we obtain
i
^(B» + bj d» ^b
(20 MR/.)  A. /.: i. + _!A+v^ F(R)
A z A
+ 6c j (x  R) dF (x),
A" JR
where A» = d3  M + f(p2 + t?) + d^ (!> s
we obtain the equations
n f 9X« &. (d),A (d)] raMd) QA, (d)
I* tE
i=llpi_ ad
aD(p)
ÖP
"aMd)
_____ +1
¦ ad 2
 0
ad
(33)
n
i1
R^d)
Ai(d) + 1
 E(P * S0J) •0.
In the above equations, we must remember that R^{d) and A^d) are
given by equations (26), and X? is given by eauation (25). The function
D(P) is derived in sections 13 and IS for various procurement policies.
Eouations (33) were obtained in the manner jhown in order to retain the
shadowprice d In a more explicit form. Actually, the shadowprice »as
used only as a heuristic device and can be readily eliminated in a mathematical
solution. Such a method might be as follows: Eliminating d. from
equations (26) we obtain:
(33a) ^.(A ,R.) = 2Bi +
i i 
+ (L
There are actually n such
values for optimal A and
We are now interested in
f C1 (x  R.WAx) + A. jf (x  R^dF^x)
eqsolving
the equation
UP^R.) «D(P) 4 ^»"i
subject to the restrictions of the equations ^(^V " 0 and the eqUati0n .
n
(^^»R^P) " I &
We form the function
AA.^P) = L(P,
where ^. and \i are Lagraif;
The partial derivatives
ÖP dP
dh pi aR.
31
i pi
Eliminating the Ugrangian m
— 'n irinurr
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49
oiD^FiCR^ViCRiQ
m. R.
=« o
lions, one for each base, each with different ^ ^
n xVVAJ
i1 Pi
A. + 1
^ + 2. VE(P * so J}  0
. ,\) + ^^i^i^i^ +^(\iRi.p)
igian multipliers (i =» 1 to n).
of P ' set to zero are
axVV'i) ^i'V + + ^i
dR.
ax'ilRL^i) ^(W + M.
+ t*  o
1 dAi
jltipliers, we get the equations
* Et
'" ax« ötf.
i i .^i axr
Pi ^i a^ aAi ^i
P6Ar'
50
ap
1 i
2 3^
(33b)  0,
which must be solved simultaneously with the equations ^P.vA.,R./ m 0 and
if(A.,R.,P) ¦ 0 to obtain values for P,Aj,, and R..
Since i *  ^i
aA. A2
1 1
m s.
(x  R^dF^x)
R.5
00
^ l&oA^i^  ci J_ i (x  %) dFi(x)],
aR. L Pit) w n
we see that
A.
i
or
f.(A.,R.)  
i i i
ax« n a;..» i  ? i
aRi"Ri
2Ai ax» A. ax»
i+i 1 = 0
i i i i
0.
OR. 8A. axj .
Substituting Q^[ ¦ 1 _! in equation (33b), we get
5^, 2 8R.
i i
2aRi ^i L Pi «i ap
o.
It can be shown that
! a^.a^ .^ LVi^)b ;f(R.)].
2 aR, cTA, ^ • ^ i i oi i i
Since this term cannot be zero for all R, we get
Et Jajx». + aD(p) ¦ o.
pi aR. ap
The system solution is then expressed by the following set of equations:
axj
(33c)
Et 1 » 0D(P)  0
P. aR, ap
i i
a 1  2 9A1  0.
an. aA.
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51
l (FL+!i) EP ^ S (1fJ + E(iS0) « 0
i«l 2 i=l 2 1
If we now wish to reintroduce the shadow price d we observe, from
equation (6), that
ÖRi ^i ^.
ax» Substituting this into our result for uy'i we see that
ax i
d.
i
which must hold for all bases. Therefore, the first of equations (33c) becomes
 JL a)(P). «¦ d.
p. Et a P
SXi
The first of equations (26), when combined with our result for 1, shows that
i
so that the second of equations (33c) is no longer independent. However, in
the third of eouations (33c.), A and R. may be expressed as functions of
JL
d^ by equations (26), Therefore, we obtain the eouations
f33d)
m
n
h
i»l
{R.UO
Et
L )  EP * T. (1  6J ^ ECi S )
? / i=l 7
which may be solved simultaneously for d. an i P, where R^d^ and
A(d^) are given by equations (* }.
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5?
Equations (33d) represent an easier set of eouations than equations (33)
for finding the optimal procurement amount, P.. A computational method might
be to assume a value for P, find the corresponding values for d from the
first of equations (33d), and calculate optimal A. and R^ for all bases,
using this value for d* in eouations (?6), When these optimal values are
placed into the second of equations (33d), together with the assumed value
for P, a result is obtained which may or may not be zero. If different
from zero, the procedure is applied for different values of P until a P^
is obtained such that the result is nearest zero. This P^ represents the
optimal procurement amounto
Eouations (33d) provide a value for the procurement amount P, which is
"optimal" in the sense that, given E, it provides least expected system
costs. Of course, there may be some particular E which .Helds lower costs
than any other value for E. On the other hand, we recognize the usefulness
of assigning E to conform with military policies. In a sense, then, we
obtain a suboptimization for the system reouirements calculation.
12. BASE STOCKAGE DISTRIBUTIONS
Having determined the amount to be procured, we now look at the problem
of distributing the materiel among the bases and depots. Of the several ways
in which this problem can be approached, we consider a method of recalculating
base stockage levels at a number of times throughout the procurement
period.
Let the procurement period oe divided into m intervals by the points
ti, to, ... t +l, where t, is the beginning of the procurement period and
t +, the end. At each time of calculation, t, (i = 1 to m), we establish
the ratio,
¦
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53
n A.(d) + 1
I CR.(d)  £". + J_ 1
(34) E  .1=1 J i ?
Si ~ J'i
where the numerator is the expected base stockage at the end of the procurement
period, S. is the total system stocks at t., the time of calculation,
i. is the expected consumption during the rest of the procurement period,
and E is the same constant used in the previous section« Various values
for the constant J!. will be given in section 13. Eouation (34) f^ay be
solved uniquely for d. This provides values for R. and A (j = 1 to n
•J J
for n bases) and determines the base stockages.
The distribution of amounts procured to the various depots is determined
at time t,, The procurement amount is distributed so that the resulting
depot stocks are in the same ratio as the expected mean demands of the bases
normally served by the respective depots. The mean demand, of course, must
be determined in terms of some common unit of time. If there are two depots
A and B, for example, then the respective stocks S^ and Sn after distribution
of the procurement amounts should satisfy the relation
_A „ J
M M '
A 3
where H, and Mn are the expected mean demands of bases served by depots
A D
A and B respectively.
Let us new look at the effect of the stockage distribution method described
above. If part way through the procurement period, the demand for
the item has been greater than anticipated, the expected system stocks at
the end of the procurement period, SJ ~ t \f W"iH decline. Therefore, in
order to maintain the ratio E, the shadow price d will increase. Thus,
all base reorder levels and stock control levels will decrease and a uniforrr
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stringency will be imposed on all bases« Base stocks will reach the reorder
levels later and when they do. smaller orders will be sent. By thinning out
stocks at all bases, this procedure reduces the probability of having to
baekorder any of them.
If, on the other hand, the demind for the item has been less than anticipated,
the expected system stocks at the end of the procurement period will
rise, the shadow price will fall, and all reorder levels and stock control
levels will rise. Thus, base stocks will reach the reorder levels sooner
and larger orders will be sent. In this way, the extra amounts will be
spread throughout the system»
13. THE PROCUREMT COST FUNCTIONS FOR EXPENDABLE, NOWRECOVERABLE ITEMS
In this section we consider different procurement cost functions which
correspond to the different kinds of procurement policies for nonrecoverable
items. In this section, we will express the cost functions in terms of S
rather than in the procurement amount P. Since S and P differ by the
constant, S0, this becomes just a matter of convenience.
First, we will define some of the terms to be commonly used in this
section.
(l) K ¦ fixed cost of procurement, including contracting costs and
s
all procurement costs which do not vary with S. For open
contract procurement, this cost is the fixed paperwork
cost associated with placing an order on the factory.
(?) U(P) ¦ unit procurement cost of the items on a delivered basis.
This function includes factory to depot packaging, inspection,
and transportation costs. When P •• 0, U(0)
is the initial setup cost3 of manufacture.
(3) c • scrap value of the items or the salvage value, whichever is
¦ . higher.
¦¦¦¦ ¦¦¦ ¦¦¦¦ 
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55
(6) f.(x)
i
(4) p •" routine pipeline tjjne, for open contract procurement. It
is defined as the number of days from initiation of an order
on the factory to the time of arrival of the materiel in the
depot. It differs from lead time, as previously defined, in
that it contains no contracting time and usually little or
no setup time at the factory.
(5) n m expedited pipeline time for the relevant procurement. Its
^s
definition is the same as for p but uses premium communication
and transportation.
system demand probability functions where different values
of i (i * li 2, 3) refer to the different procurement
policies. Thus, f, (x) represents the demand probability
for the remaining life of the item as used in the lifeoftype
procurement policy, f(x) represents the demand
probability over the consumption period as used in periodic
procurement, and f„(x) represents the demand probability
per p, the routine pipeline time from the manufacturer
to the depot. This last function is used in the open contract
procurement policy,
x
(7) F.(x) ¦/ f, (t)dt ¦ cumulative system demand probability function
o
(i ^ 1, 2, 3») corresponding to the threeprocurement
policies,
(8) £. ¦  xf.(x)dx ¦ expected mean demand (i ¦ 1, ?, 3) correspond
•^o
ing to the three procurement policies,
(9) ,dj . 0e.y ¦ cost of holding an average amount y, where the sufescript
i (i"l, ~, 3) refers to the three procurement
policies. Thus, when i " 1, this cost represents the
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56
holding cost over the remaining life of the item,
Wien 1 ¦ 2, it is the holding cost over the consumption
periods and vtfien i ¦ 3« it is the holding
cost per p . These costs include interest, warehousing,
and all depreciation costs except obso ,
lescence.
(10) b + b..q + b p q ,B system depletion penalty, or the cost of
SO Sx S^S'
insufficient procurement. In this cost^
b is the fixed cost,if any, which is so i * *
independent of the amount of depletion«
The cost b. is the cost associated with
s i
the amount of depletion a, but not duration
of depletion. The component sbo„^psq
is the cost associated with the amount and
duration of depletion. Here, b? could
be the ooat per consumption period of having
an extra end item available for use. The
term , p is the duration of depletion, in
terms of a fraction of the consumption period,
while q is the amount of depletion. The
various components of the depletion penalty
will be discussed in further detail later.
A» LifeOfType Procurement
The procurement cost function, D(S) for a lifeoftype procurement
policy and for a nonrecoverable item is given by
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57
(35) D(S)  Ks + ü(OJ + [ü{S  30)  el] I '(¦S  x) dF, (x)
0
i.  1
+ d + e ( 3  ) * Qj(0) + K "1 &  Fi (3)1 *
The second tenn rU(S  3 )  cj I (3  x) dP,(x), in the right member
of equation (35)j represents the expected obsolescence cost of procuring
P items. In this term, we assess obsolescence of an item procured in accordance
with the probability of that item not being used. If x items are demanded
during the life of the item, and there are S items in the system
initially, then S  x items are not demanded, (x < S). We multiply this
number of items by the unit cost of the item, [u(S  S )  cj and by the
probability of demanding x items dF (x). We then sum, or integrate, for
values of x ranging from 0 to S , where the symbol S means integration
to S but not including S. It might be noticed that we are pricing the
amounts So which are on hand at the time of calculation at the »ame unit
cost as the amounts procured.
The terra S  in equation (35) represents the average expected
amount on hand during the life of the item. Its derivation is similar to
the derivation of equation (3). This term is multiplied by e, the component
of the lifetime holding costs associated with the amount held.
The last term of equation (35) represents the depletion penalty, where
U(0) + K is the cost of depletion and 1  ?A3) is the probability of depletion.
The depletion penalty expressed in this way assumes that the depletion
iä predictable sufficiently in advance to obtain more materiel from the
factory before actual depletion occurs. Therefore, we limit depletion costs
to the fixed costs of procur^ment plus initial setup costs at the factory.
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If such additional procurement is made, then equation (35) is used again to
find the amount of procurement« Of couse^ the function F, (x) will be
different in this case, since the expected remaining lifetime of the item
has changed.
The value of £ for use in equation (33) is £,. To find the value of
£. to use in equation (34) we suppose that £,. is given for i ¦« 1 to m,
where £,. is the expected average system demands during the ith calculation
period, and there are m such periods in the expected life of the item,
n m
If this is the case, ^i a t m ^ ^li an^ ^ " ^ ^]ke **" ^^e exPec^0d
i8»! koi
demand during any one calculation period is assumed the same as for all
t1(mi)
others, then £. * , The value for t for use in equations (33)
m
is the expected life of the item, expressed in days.
B. Periodic Procurement
The procurement cost function D(S) for a periodic procurement policy
and for a nonrecoverable item is given by
s"" r
(36) D(S)  K + U(0) + ru(S  Sj  cj / (S  x) dF1 (x) s " ^0 1
+ ,do + ^o & '  ) + S / S > r sbo fl  ?*J3) I
\J (x  S) dF0(x)  b, J (x  S) (x H 1  S) dFJx).
i s / s ' s ? (x + 1) ä
This equation is similar to equation (35) except for the last three terms
which represent the depletion penalty. The term hJjL  F (S)J represents
the costs of depletion which are invariant with the quantity of depletion and
the length of iepletion, where gb is the cost and \\  F0(S)]] the probability
of incurring the cost. The term b, J (x  S) dF?(x} represents
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59
the costs of depletion which vary with the amount, of depletion but not with
the length of depletion., In this term b is the cost per depletion^
s X
x  S is the.amount of depletion and dF (x) is the probability of demanding
x items and incurring a depletion of x  S items. The integration sums^
for values of x from S to oo, the products of amounts of depletion by
probability of incurring that amount of depletion.
The term b f (x  S) (x .• 1  S) dF (x) represents the costs
s ? JS 2 (x + 1) ^
of depletion which vary with the quantity of depletion and the length of
depletion. In this term, b is the per unit cost of such depletion,
s c
(x  S) is the amount of depletion, x + 1  S is the expected duration
"? (x + 1)
of depletion, and dF (x) is the probability of depletion by such an amount.
These expected costs for all possible depletion amounts are summed by integrating
over values of x from S to oo.
The depletion penalty defined above assumes that the item cannot be recontracted
for before the established contracting date if the system prematurely
runs out of the item. However, if we do assume that additional
quantities may be contracted for before the regular contracting date, then
the depletion penalty becomes LK + U(0)J \j.  Fo(S)]], as was the case
s *
in lifeoftype procurement. This assumes, of course, that the system is
able to predict the impending depletion and obtain more materiel before
actual depletion occurs.
The value for £ for use in equations (33) is £». As in the case of
lifeoftype procurement, the value for £* to use in equation (34) is given
n
by i tt £ L,, where ^0. is the expected mean demand during the kth
i k^^K
calculation period, there being n such periods in the procurement period.
The value of t for use in equations (33) is the procurement period, expressed
in days.
«;¦¦
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~6Q~
C. Open Contract and Variable Period Procurement
For the open contract procurement policy, we have the procurement cost
I <
function expressed in terms of two variables rather than the single variable
as in the other policies« These two variables are the system reorder level
R_ and the system reorder amount (amount procured), A. The procurement
cost equation is very similar to equation (5) for the base costSe The same
terminology and definitions will therefore be used but prefexing s as an
index to the various costs to distinguish clearly the two cases. The procurement
cost function for open contract procurement of a nonrecoverable item
then becomes
(37)
^(B + b ) e0
s
CO
A s ^Aa ns
where A  d + £ ( p . t ) + e (1 ~ £ )
$ s3 3s? 3 2 s 3 ? 3
B «* K + p,+ t,
S S Srl 3 1
C =» b + b0n
3 8 1 3 ?HS
To solve for optimal R and A we must develop equations to replace
s s
equations (33)c An eouation analogous to equation (30) may be expressed as
follows:
(38) L(Rs,Vd ) > D(R8»AB) + \s(t )'
This equation is subject to the restriction of equation (?9), whe
' '' •,
re
sBas^
Using the method of section 11, we form the function
¦ ¦ — '  • I
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61
and solve the following set of equations simultaneously for ^,4,%, and ^j
a/7 « öD_  0
hP " öD  iE ° 0
oZ:^ iMi K.aa^^ ^ ^ 0
a^ p1 d^
a/7 = t a^^
dÄi Pi a"i
^i(AilRi) » 0
WA +
ö^ 2
Using the results of section 11, we obtain the equations
(39)
Pi aD(SR^^SJ'
Et OR.
n
i1
\{\) W
EH. + I (i 6) * E(,^ 3 ) « 0
0 i«l ? 0
ao » 0
which may be solved simultaneously for d., R , and A • Of course, R.(d. )
and A. (d.) are given by equations (?6).
The value for £ in equations (39) is £ . The value for t in equations
(39) is p • It is noticed that the process of section 12 is not particularly
applicable in the case of open contract procurement. The value for d which
determines base stockage is obtained from equations (3°), and does not change
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«62
unless from a recalculation of equations (39).
Equation (3?) tioes not contain obsolescence costs as did the equations
for the other procurement policies. This is under the assumption that the
open contract policy would not be used when the item approaches the end of
its life* However, if we are willing to recalculate optimal R and ^ 5 8
each time the reorder level is reached, we may introduce obsolescence byadding
the following terra to equation (37)J
R + i
UJKAJCI fs «
J (Rs ^5  x) dF (x)
A« 0
s
which is as previously defined except for dividing by fs. * Ö to express
the cost per routine pipeline time, p .
s
f s
H. THE (SHSRAL SYSTSt^l SOLUTION FOR RECQVERABIJ: ITEMS
The general system solution for recoverable items is similar in concept
to the solution for nonrecoverable items presented in section 11. Modifications
must be made, however, to allow for items being in repair and hence unavailable
for supply. Also, it is necessary to allow for gains to the system
from attritted end items and for losses due to condemnations.
Before introducing the system cost equation for recoverable items, further
notation must be developed. This notation will also be used in section 15.
(l) t « length of the consumption period, expressed as a number of units
of time. The unit of time must be significantly large; in the
order of a month. If the unit of time is a month, then there
are t months in the consumption period.
(?) t» ¦ the remaining lifetime of the item, measured in the unit of time,
(3) ^J ¦ base stockage cost for the ith base (i1 to n) during the 1th
P1 unit of time (j ^ 1 to t), where p is the routine pipeline
time of the ith base exoressed in terms of the unit of time.
.... iriiiTiiinMri
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(4) Jj • expected exce33 transportation costs dnrlng the Jth unit
of time (j=l to t),
(5) r. "^ repair cycle ending with the jth unit of time (j»l to t)»
Here, r. is expressed in terras of the unit of time«
(6) f (x) « system deimnd probability function over repair cycle rr
x
(7) F/.<(x) ^ / £1 *Mtä m cumulative system demand probability
function over repair cycle r
(8) £. " mean expected system demands during the jth unit of time
4j
(not the jth repair cycle).
(9) w * average wearout rate, or the ratio of items condemned to
total exchanged items. It is used below as the probability
of an item received in exchange being condemned.
(10) w « number of items condemned; a random variable.
(11) (1)J(W) c the condemnation probability function for time periods
«j
from the time of computation up to but not including repair
cycle r..
¦ I 4,(x,w) dP.(x), where ^(x,w) is the probability of
condemning w items when x items are demanded (w < x)j
and F.(x) is the cumulative demand probability function
for time periods from the time of computation up to but
not including repair cycle Ty It can be shown that if
x takes only integral values, then (j/xjw) « (1  vr)
w t 1/ w xl
wl (x  w)l
1/ See J. V. Uspensky, INTRODUCTION TO MATHEMATICAL PROBABILITY, p 46.
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w
(12) ü).(w) =' J (j).(t)d.t » cumulative condemnation probability
3 o ^ ,
function for time periods from the
time of computation up to but not
op Including repair cycle r.«
(13) (JKw) * J ^(xjw) rdF1 (x) = the lifetime condemnation probw
f
w ability function. r
(14) 2)(vr) "J (ji(t)dt  cumulative lifetime condemnation prob
0
ability function,
(15) (D?(w) =» cumuLative condemnation probability function for the
consumption period.
(16) a. » expected total number of items (assumed reparable) obtained
from end items which are attritted in the jth unit of time.
It does not include condemned items from attritted end items«
Each a.j. is assumed constant.
(17) S* ^ S + Z a, ,0 expected amount of serviceables, reparables
J ' k«l K
and condemned items in the system at the
beginning of the jth repair cycle.
(18) v(S) a average expected value of an item in a reparable condition.
This value may be given as the procurement unit cost "0(3),
minus average cost of repairing the items minus average
transportation costs from base to depot. The value
v(S)  c is always positive since if the scrap value is
greater than v, the item is condemned rather than repaired.
(19) p* » procurement and contracting lead time measured in units of
time.
An eouation analogous to equation (?ß) may now be expressed for the case
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of the recoverable type item as follows:
*] r
3 t rJ f y*
(40) L(P)  D(P) +2 J / M^P^w) dP, .(x) d (D,(w)
j«p»+l wo x=0 ,J 4J J
+ Z J M.CP.xSJw^) QFM(S'w)D dffl.Cw)
t
+ E M.(P,XO,^S;) Q.  QLCSOI
jpt+i J j J j 
vrfiere n Xf.(P#x,w)
M.{P,x,w) « S t Jj + J.(P,x,w)
3 1=1 p. J
Equation (40) differs from equation (28) by the addition of the random
variables, w and x, which represent the amount condemned and the amount in
repair, respectively. The function M. is only implicitly a .function of
P,x, and w; in particular, X?. is explicitly a function of Ri and £. .
Of course, equation (40) is restricted by the condition that total expected
base stocks do not exceed total expected system stocks. This restriction
is expressed by equation (41) below, where E is set equal to 1. Different
functions for D(P) in equation (40) are developed in section 15 to correspond
to the different procurement policies.
In equation (40), the functions J, are impossible to obtain, so we
again resort to a suboptimisation by use of a fixed ratio of base stocks to
system stock. Here, however, we impose this ratio for each unit of time in
the consumption period. Therefore, we obtain the equations V1
(41) E  E  W
zLhi *   hß
S»  x  w
J
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^y a method analogous to that of section 11, we obtain the optimal
procurement amount, P, as a solution of the equation;
(42) E J J j d (P,x,w) dP (x) d Q)(w)
s
£ / ^ d ,(P,xSV w>w) (JLF (SJw)] dffl.Cw)
1»n»+l T^O lJ J 4J J J
3
p * I d . (P,x»0,vP=St)Clffi (SOI  ^i dD(P)
j»pUl iJ J J J tE öP
where d ,(P,x,w) are given as so3.utions of the eouatlons
iJ
n
i1
A (d ) n
+ E (1  6. )  E.S! + E,(x*w) * 0.
i«l 2 ¦j' J J J
Of course, in the above equations, R^fa 4) ^d u^ .(d. .) are solution
of equations (8) and (9) expressed in terms of the ith base and jth period.
Also, it should be remembered bhat SJP+S . E^a, .
The above results are used only bo find the procurement amount P, and
are not used for subsequent base stockage distributions. The discussion of
section 12 still applies for the base stockage distributions, except that
equation (41) is used instead of eouation (34).
15. THE PROCUREMEMT COST FUNCTIONS FOR EXPENDABLE. RECOVERABLE ITEMS
In this section, we consider different procurement cost functions which
correspond to the different kinds of procurement policies for recoverable
items«
A • Life~ofT.vpe Procurement
The procurement cost function D(S) for a lifenftype procurement policy
^esttääääägäeäoi rraiiiiisrnMafc rrri ^^aüaiiMiiaiabMa*
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and for a recoverable item is given by
(43) D(S) Kg * U(0) 4. Cv(SSo) ~ c]
t»
(1  w I
kttr. ?+l
'4k
a d + e i S + k=l
si si!
t» _ t»
£ ak  w £ £, v
+ 1
k»l 4k
?
(K + ü(0)) 1 (D(S + E a,  (1 w) 2 £,k)
k«l K ktr +1
+ b z:
s»
^3 1  F. .(s;w) d (D (w)
J
+ b E
31 jP»n
s«
J
,00
j
(xS^w) dP4j(x) d !D (w)
S ^ Jp»+l 0
1 .1 00
" / (xSJ+w) (x+lS!+w)
J
3»w
j
?(>:+! f
^ dF4j(x) dffi M
t» t»
where S ¦ 3 ¦.• 2 ai.  E £ 4].
1 k1 ktr +1
In equation (43) the second term in the right hand side of the equation:
0(35J  cD
t»
(1w) E
k
...St^
i!rt.+l ^k J L
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represents the cost of obsolescence. In this term» [Iv(S"S )  cJ represents
the cost of one item becoming obsolete. By using QvCSS )  cj rather than
[]u(SS )  cJ we assume that the system is able to predict the remaining demand,
at some point in time close to the end of the life of the item, sufficiently
well to halt repair on items that are not going to be issued« Thus, the
system saves the cost of repairing and transporting the reparable which is
never going to be issued. We feel that if [jKSS )  cj is not an exact
statement of the loss due to obsolescence on a recoverable item, it is at
any rate preferable to Qj(SS )  cj#
_ t*
The term (lw) t £.u represents the number of items that must
kt»rtHl k
become obsolete. It is the number expected to be issued in the last repair
cycle, minus those that will be condemned in that time period. This amount
must become obsolete'even if it is necessary to procure more of the item to
compensate.
The term S represents the amount of stock that must be condemned in
order to avoid all obsolescence on the item, except that amount which cannot'
be avoided, as.described in the preceding paragraph.
S
The term f (S. ~w) d ffl(w) represents the expected number of i V
items becoming, obsolete. The term S w represents the number of items
becoming obsolete, given w condemnations; dfl) (w) is the probability of
w condemnations in the lifetime of the item, and the product (S w) d!It(w)
represents the product of the number of items becoming obsolete, and the
probability of incurring that number of condemnations. The integral sums
these products frora w » 0, in which case the whole of 3 becomes obsolete,
to S. , , in which case none oi' S becomes obsolete,
t« » t»
The fifth term in the right hand side of the equationJ
s 1
tt t»
E a. « w E £, + 1
S + kl K lc«l ^
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represents the holding costs, excluding obsolescenpej where the bracketed
part is the average expected amount of the item on hand over the lifetime
of the item.
The sixth term:
(K + U(0))
9
t t
1  (D ( S + Z &, ~ (1w) Z
k1 K kHr++l 4k
t
)
represents the expected cost of having to procure a second time. The terra
(K + U(0)) is the cost of having to procure a second timej
5
1  (D (s + E ak  (1w) t E )
1
k*l k*tr. +1
t
t _ t
represents the probability of condemning more than S ¦«• £ a,.  (1w) E f
k1 k ktrt+l 4k
which is the maximum amount of stock that can be condemned without an additional
procurement of stock. In assessing this cost, it is again assumed that additional
procurement can be obtained before actual depletion of serviceables and reparables
occurs.
The seventh terra:
t N
b l J [lF (St  w)] d fflfw)
' s
represents the expected system depletion costs which do not vary with either
the size of depletion or the length of depletion, resulting from a temporary
pileUD of reparable items undergoing repair. The terra b represents the
s o
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cost of one such depletion; the term Tl  P. .(S*w)j represents the
4j j
probability of exchanging more than S!w during the repair cycle r.;
ds ^ average system procurement period, and is defined as the r
average period between successive deliveries to the system.
(5) S" « R + A + E J a.  w Z J £,.
(6) All other terms are as previously defined.
The total system cost eauation, analogous to equation (40), may now be
exnressed as follows:
P647
^ + 1
(45) ' L(Ra,^) =D(Rg])As) + ^ ^S M (Rs^,x) dP4j(x)
&s + 1
n
where •j\"g»asJ
Ecuation (45) may be solved for optimal R and A by setting the S 3
partial derivatives with respect to R and A equal to zero, and solving S 8
the two equations simultaneously for Rs and *, , again with the restraint
that expected base stocks do not exceed total system stocks.
Again, as in equation (41) we suboptimize by replacing the J. functions
with the restriction that R. and t satisfy the equations 1J iJ
E = E. B i'l
I l\ 1J
.+1
£iP
S»T~x
J
We may now write the procurement cost function D(Rs .& s ) as follows
(46) Dft.A ) 
£»K
s' s'
s + d0 . e„ s 3 s 3
R + S  £«
9 
£« (P, + st2) + ^ (3?, + 3^)
6« b £»
+1 +1 ^P
3 0 E ["I F, ,(3")]+ 5» b Z J (xS")dP (x)
.0 4j .1 " si is7 s" J 4J
r, b ^ H f6 (xS")(x+lS")
4j
s J?
J
T^T)
In equation (46) the first four terms are similar to the same terms in
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equation (37) the last three terms are similar to the three depletion terms
in equation (44), «xpept that condemnations are treated as an expected mean
value rather than as a random variable. We "were able to make this simiplification
in equation (46) because of the fact that the system reorder level
is defined in terms of an amount of uncondemned items, and therefore, constrains
the variation in the system stock of uncondemned items within limits«
Thus, we may expect S'J to attain each value between R + A  w £? and
' j s s
R once each ordering cycle, and may expect, by averaging over a large
s •
\ number of cycles, that S" will exist for a period of time equal to that
which would be obtained by using an expected value for condemnations. We
also assume that p is shorter than the repair cycle, so that any demand
which occurs after R_ is reached will lead to a reduction in the stock of
 serviceables which is independent of whether the item received in exchange
'is condemned or reparable. ^'^¦a!a^^^immmm^mx^i*pm^m^'
Note that the summation in the depletion terras is from ? to Ö3+ 1.
The time lag of one period represents the procurement pipeline ps ; ps is
added to both the lower limit and the upper limit of the depletion term
summations since the R and A to be calculated cannot affect the probs
s
ability of depletion during the first period, but will affect, and indeed
A
.will determine the probability of depletion in period Ö5 + 1 (or u^ + 1).
If we are willing to recompute R and & each time the system reorder
o s
level, R is reached, we may add an obsolescence term to equation (4o)j
s •
r t» on
+ [v(SS )  c] (1w) Z f4k + f (ow) dai(w)
L ktir, t»l ^o J
t»
where 0" H +" iat1 t c, ^
s s ktt~rtt+l
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16. FIJKDS AND BUDGET
In computing the impact of the procurement requirements calculations on
either available funds or budget requirements, the calculation is carried
forth for as many periods t or Q as are necessary to exhaust either
the operating or the budget period. In the lifeoftype calculations, the
procurement amount must be multiplied by the unit cost to obtain the funds
or budget impact. In the periodic procurement calculation, the procurement
amount must be calculated for as many procurement periods as are wholly or
partly in the operating or budget period. In each calculation, it is
necessary to estimate the balance on hand at the time of the next calculation.
tpt
This amount is 3* f ~ w £ £.. . In calculating the procurement requiret
pl
ments for the next period, S? . ~ w £ £ is used as the beginning
ps k1 ^
balance on hand. The total amount to be procured is then extended by the
unit cost of procurement to obtain the impact on funds or budget requirements,
For the opencontract or variable period calculations, the impact on
funds or budget "requirements is R + n A ~ S , where n is the number of
periods 0 wholly or partly in the operating or budget peri d, and S is
the current balance on hand.
If there is a fund limit within which all procurement requirements must
fit, it is possible to suboptimize for procurement of all items subject to
the fund limit. Clearly, the fund limit must be considered a ceiling only;
that is, if the dollar requirement for procurement of all Items is less than
the fund limit, it cannot be considered as either optimal or suboptimal to
raise procurement up to this limit. The problem before us 'consists of the
necessity of forcing the dollar procurement requirement down to the fund
limit, or, in other words, to suboptimize procurement under constraint of
the fund limit.
a».—iii IM an—a mummi
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For this purpose, we may add the terra riU(S)S to all lifeoftype and
periodic procurement equations; and the term l2ü(^ ) (R + A" ) to all
s ^
open contract and variable procurement equations. The parameter jQ shquld
be held at zero for the first calculation of the dollar requirement for procurement.
If the total dollar procurement requirement is greater than the
fund limit, fl may then be raised in increments until the dollar procurement
requirement is equal to the fund limit.
17. REFINEMENTS TO THE SYSTEM SOLUTION
A. The term b as a Function of Amount of Depletion
In all previous formulations b0, the coefficient of the cost of
system depletion which varies with the quantity of depletion and the duration
of depletion, has been assumed to be constant. We may wish to make b a
s 2
function of the amount of depletion. For example, the military organization
may be able to prevent any loss of time in commission of the applicable end
item, despite a depletion, through the device of "maintenance cannibalization";
in which the depleted item is removed from an end item entering maintenance
and placed on an enditem which only needs the depleted part to become available
for use. In this way, an end item out of commission is avoided by incurring
the cost of one extra removal and one extra installation of the depleted
spare item. It would not be appropriate to use this cost of an extra
exchange as the depletion penalty since maintenance cannibalization is probably
limited; it may suffice to prevent end items out of commission for the
first ten depletions, but beyond that, depletions may cause end items to be
out of commission. Therefore, b^ becomes a function of the size of des
2
pletion.
We may introduce this modification into the system equations merely by
moving b inside the integral as a function of the amount of depletion
awaaa——wni
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whenever it appears; that is, in equations (36), (37), (43)* (44)» and (46)«
For example, in equation (36), the last terra would become:
00
b0 (xS) (xS)(x<lS) dF0(x)
s 2
>(x+l)
Here, b (xS) is the cost, which varies with the amount and duration
5 2
of depletion, of (xS) depletion?.
The same modification can be performed on bp in the base equations.
B* The Obsolescence Concept
Our treatment of obsolescence in the system equations is based upon the
assumption that all end items will be phasedout as whole end items; that is,
not lacking the spare item. Thus, the number of spare items condemned represents
a fixed minimum requirement for the spare item. The logistics system
is not charged the procurement cost of the condemned item since it is assumed
not to be within its discretion whether that item is bought or not. It is,
however, charged a holding cost if the item is bought too early, and a depletion
cost if the item is bought too late. The assumption will be completely
valid if, for example, the military organization wishes to mothball the
applicable end item as a whole end item at the time of phaseout; or to present
the end item to a friendly nation, again, as a whole end item.
If the military organization is willing to phase izs end items out as
incomplete end items, we may substitute the term:
U(S)S
for the obsolescence term in all system equations.
C. Variable End Item Maintenance Flow Time
The terms b^ and gb^, whether constant coefficients, or variables ,
have been developed on the assumption that the maintenance flow time of the
end item is given. If b0 and b0 are interpreted as the loss of utility
P.647
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of the end item, they must be modified by the expectation that the end item
requiring the part may have been out of commission already when the demand
for the spare item arose. The end item may have been out of commission for
another part, or, more likely, may have been out of commission because the
need arose while the end item was undergoing overhaul. If, on the average,
three depletions out of ten lead to end items out of commission and seven
out of ten do not,'for either of the foregoing reasons, b and ^b must
be developed as .3 times the lost utility of the end item, rather than the
whole lost utility of the end item. If the maintenance flow time of the
end item were reduced, as a matter of policy, then we should expect that a
larger per cent of depletions would lead to end items out of commission.
If maintenance cannibalization is used to measure the depletion penalty,
the maintenance flow time of the end item becomes important in determining
the number of exchanges in the spare part repair period of time in which
bp is defined. For example, if the spare part repair period of time is 30
days, and the end item maintenance flow time is 10 days, a system depletion
of the spare part lasting one period of time will be valued at the cost of
three exchanges through cannibalization. If the maintenance flow time for
the end item is then cut to 5 days, the same system depletion of the spare
part will be valued at the cost of six exchanges.
Thus, b9 and b^, and also L, the system cost of logistics support
occasioned by one spare item, become functions of the end item maintenance
flow time.
Let us introduce some new terminology at this time:
(1) m. ¦ maintenance flow time of the applicable end item in period j.
(2) L.(ra.)  lifetime system logistics cost of spare item i as a function
of ra^.
mamaaesm
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80
(3) t « total useful life of the end item.
(4) M (ra.) •• lifetime cost of a pool of end items out of commission for
j
maintenance as a function of m..
(5) M^m.)  cost of maintenance on the end item in period j, including
direct and indirect labor and capital, as a function of m.»
(6) C* ¦' total lifetime cost of maintenance, logistics support, and a pool
of end items out of commission for both maintenance and parts,
over the lifetime of the end item.
We may now write an equation for C. as:
n .1. x « „2.
(47) C. (m.)  I L.(m.) + M (mj * I M^m )
t J i«l 1 J J j^l j
where there are n spare items applicable to the end item.
We may then take t partial derivatives', one for each of the m.j set
them equal to zero; and solve simultaneously for the m..
D. The Batching Concept in Repair
In defining the repair cycle in period j as a constant, we have assumed
that all reparable items are scheduled through repair as fast as they are generated
(although the repair cycle may vary between periods). The military
organization, however, may be faced with a large fixed cost per batch of
items scheduled through repair. We present below an alternative formulation
of the system cost equation which may be used for determining simultaneously
the requirements from procurement and the optimal batch size for repair. We
shall present' the case of lifeoftype procurement as an example.
Lot us .first introduce some new terminology:
(1) t» ¦ the lifetime of the item measured in major periois of time
(periods in the order of a year) t .
m
P6/x7
(6) S A
(2) t ra the major period of time measured in minor periods of time, p .
R
(3) p « the minor period of time which is defined as the period from
starting the batch of reparables through repair to their emergence
as serviceables.
(4) A_ » the repair quantity or batch size in the jth major period» (^ > 2_\
HJ Rj '
(5) 3 w = S + E av + Ü  w " the average ouantity of serviceables
RJ k=l 2
and reparables in the system in the
jth major period (assumed constant
over the major period), whore w, as
before, represents condemnations and is
a random variable.
w « the repair level, or quantity of serviceables in the
system at the time that repair on the batch is oegun.
(7) fn (x) " the demand probability function over the minor period Pp
Rj "
in the jth major period.
(8) FD.(x) « ( fn,(t)dt ¦ the cumulative demand probability function
over minor period p in the jth major period.
(9) ÖD4 "' " the average cycle between receipts from repair in the jth
3 major period, where £pA is the mean of f (x).
(10) (D.(w) » the cumulative condemnation probability function for periods
prior to major period j, and for half of major period j.
(11.) p. » the routine depottobase pipeline for base i, measured in units
Of pR.
(1?) 1 . d Ay) ¦ the holding cost over the major period, where y is
Rl R«?
the average amount of the item in the system in the
major period
aaaisgBacMBtt
TSltf
i 82
(13) Vf * Vyiq) = the cost"of repairing a batch of items, where q is
the number of items in the batch
(14) J.(SD..x,w) ¦ excess transportation costs defined over the minor
period, p„.
An equation analogous to equation (40) may now be expressed as follows:
•' " '^^ I . 'XT !
+ rRj VVV^ Cl.F(x0dx
00 (x~S ,+A
r n.+w) . v / .] gj M.(SD.,x»S z^.Wjw) dF(x)ida)(w),
S ^ w öRj Rj
r Rj' vRj "Rj
where Mj(SR ,x,w)  Z ^ Rj + )
There are two rather questionable assumptions implicit in these equations.
First, it is assumed that the wearout rate is low enough for S« . to be a
reasonably close approxinßtion to the stock of serviceables and reparables
throughout major period j. Second, it is assumed that the probability of
x v> A is negligible. The formulation could, however, be refined, and the
first of these assumptions relieved somewhat, by reducing, through an iterative
process, the major per'^d tv, until it approaches the largest 9_.,
4