UNCLASSIFIED AD NUMBER LIMITATION CHANGES TO: FROM: AUTHORITY THIS PAGE IS UNCLASSIFIED AD224373 Approved for public release; distribution is unlimited. Document partially illegible. Distribution authorized to U.S. Gov't. agencies and their contractors; Administrative/Operational Use; MAR 1955. Other requests shall be referred to Defense Public Affairs Office, DFOISR, Washington, DC 20301. Document partially illegible. rand corp ltr, via DoD dtd 23 aug 1967 • - •li UNCLASSIFIED Ao 2 243 7 3 DEFENSE DOCUMENTATION CENTER FOR SCIENTIFIC AND TECHNICAL INFORMATION CAMERON STATION, ALEXANDRIA. VIRGINIA UNCLASSIFIED NOTICE: When government or other dravings, specifications or other data are used for any purpose other than in connection with a definitely related government procurement operation, the U. S. Government thereby incurs no responsibility, nor any obligation whatsoever; and the fact that the Government may hsve fonaulated, famished, or in any way supplied the said drawings, specifications, or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use or sell any patented invention that may in any way be related thereto. ·•· THIS DOCUMENT IS BEST QUALITY AVAILABLE. THE COPY FURNISHED TO DTIC CONTAINED A SIGNIFICANT NUMBER OF PAGES WHICH DO NOT REPRODUCE LEGIBLYo I. .r Co sii® /die/ "O n o V. >* ¦ I ¦ ¦ ¦ . - r-\ r" '-vfk MAR 3 6 1^ ; iUuJ T1SIA ß 41^ ^¦¦^.-¦^r-^-. ..¦..,¦ --.:tv.-.| mmaammmm ^an¦ —¦ Sy-fJ9 .-P^ AN OPTIMAL INV^'TQRY POLICY FOR A MILITARY ORGANIZATION Edward B. Berman Andrew J, Clark. P-647 March .30, 1955 ..- , As ¦ . -?<&mm(?ÖX/t01->- ^.- ..^-^.^ P-647 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 P-647 -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 56-6?, 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 P-647 -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 are-ised. 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^ P-647 -9- Stieltjes sense over all intervals on the positive x-axis 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, non-homogeneous 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 (b-i + 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 P-647 -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 P-64? - -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' i-th 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 i-l 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 P-647 -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+A-6 \ on Hand -•. 1 -^ 1 R --1 o Ü Tljns N'otice that the step-function character of Figure (4) is due to the fact that P-647 -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 step-function 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. = k-1 i 9 as the arithmetic mean demand for the i-th 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 i-th time the balance reaches the reorder level to the i+l-th time. Then the average stock on hand during the k-th 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' (x-R)dF(x). A A R P-647 -16- demand per pipeline time as previously defined. To show the derivation of these equations more clearly, we might isolate the k-th 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-:—"'--- ¦- ' " ' —.-. .„>..^-'. P-647 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 re-subdividing 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 i-1 n n i-1 j-l n &i (j -1) x.. n p9 E E 2 iJ - E 2 (k - i)xik pe - E ¦1 .i-1 nnft ivl lc"1 npS k-1 p9 (k = E 2 K k-1 pO pO But x, -^ I for each k as n -^ oo. Substituting, we get taa i—aaao——wwi P-647 -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 recoverable-type 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 base-reparable, it can readily be converted to a serviceable item - more readily, in fact, than obtaining the item from any other source. Hence, a base-reparable 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 P-647 -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-^ ^ / / P-647 / ' -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 - "•' "-*" P-647 ~?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 A-R plane represents those values of A and R for which the X - surface lies below the t) - plane . From - . . ^ P-647 -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 cross-hatched 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 high-cost, low demand items or for low demand items with low depletion penalties. For low-cost, 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 low-cost 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 P-647 -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 P-ö47 -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 R-o) X([A^I, R=0) XCAJG, R-l) 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.. P-647 -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 P-647 -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 ...P-647 -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 P-647 ¦ -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 i-1 R^d) Ai(d) + 1 - E(P * S0-J) •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 shadow-price d In a more explicit form. Actually, the shadow-price »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 P~647 -49- oiD^FiCR^-ViCRiQ m. -R. =« o lions, one for each base, each with different ^ ^ n xVVAJ i-1 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 P-6Ar' -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. P-647 -51- l (FL+!i) -EP ^ S (1-fJ + E(i-S0) « 0 i«l 2 i=l 2 1 If we now wish to re-introduce 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 (* }. P-647 -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 sub-optimization 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, ¦ P-647 -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"i-H 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 P-647 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, NOW-RECOVERABLE ITEMS In this section we consider different procurement cost functions which correspond to the different kinds of procurement policies for non-recoverable 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 set-up cost3 of manufacture. (3) c • scrap value of the items or the salvage value, whichever is ¦ . higher. -¦---¦-¦--¦ --¦¦¦-- ¦¦¦¦ - P-647 -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 set-up 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 life-oftype 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 P-647 -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» Life-Of-Type Procurement The procurement cost function, D(S) for a life-of-type procurement policy and for a non-recoverable item is given by P-647 -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 set-up costs at the factory. P-647 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 i-th 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(m-i) 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 non-recoverable 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 P-647 -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 life-of-type 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 life-of-type 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 k-th 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. «;¦¦ p P-647 ~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 non-recoverable 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 sB-as-^ Using the method of section 11, we form the function ¦ ¦ — ' - -• I P-647 -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 i-1 \{\) 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 P-647 «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 UJKAJ-CI 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 non-recoverable 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 i-th base (i-1 to n) during the 1-th P1 unit of time (j ^ 1 to t), where p is the routine pipeline time of the i-th base exoressed in terms of the unit of time. -.... iriiiTiiinMri P-647 (4) Jj -• expected exce33 transportation costs dnrlng the J-th unit of time (j=l to t), (5) r. "^ repair cycle ending with the j-th 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 j-th unit of time 4j (not the j-th 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. P-647 -64- 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 j-th 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 j-th 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 P-647 -65- 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.x-SJ-w^) Q-FM(S'-w)D dffl.Cw) t + E M.(P,X-O,^S;) Q. - QLCSOI j-pt+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 sub-optimisation 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 P-647 -66- ^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,x-SV w>w) (JL-F (SJ-w)] dffl.Cw) 1»n»+l T^O lJ J 4J J J 3 p * I d . (P,x»0,vP=St)Cl-ffi (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 i-J n i-1 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 i-th base and j-th period. Also, it should be remembered bhat SJ-P+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~of-T.vpe Procurement The procurement cost function D(S) for a life-nf-type procurement policy ^esttääääägäeäoi rrai-iiii-srnMafc rrri ^-^aüaiiMiiaiabMa* P-647 -67- and for a recoverable item is given by (43) D(S) Kg * U(0) 4. Cv(S-So) ~ c] t» (1 - w I k-tt-r. ?+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 k-t-r +1 + b z: s» ^3 1 - F. .(s;-w) d (D (w) J + b E 31 j-P»n s« J ,00 j (x-S^w) dP4j(x) d !D (w) S ^ J-p»+l 0 1 .1 00 " / (x-SJ+w) (x+l-S!+w) J 3»-w j ?(>:+! f ^ dF4j(x) dffi M t» t» where S ¦ 3 ¦.• 2 ai. - E £ 4]. 1 k-1 k-t-r +1 In equation (43) the second term in the right hand side of the equation: 0(3-5J - cD t» (1-w) E k ...St^ i!-rt.+l ^k J L P-647 -68- represents the cost of obsolescence. In this term» [Iv(S"-S ) - cJ represents the cost of one item becoming obsolete. By using QvCS-S ) - cj rather than []u(S-S ) - 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 [jKS-S ) - cj is not an exact statement of the loss due to obsolescence on a recoverable item, it is at any rate preferable to Qj(S-S ) - cj# _ t* The term (l-w) t £.u represents the number of items that must k-t»-rtH-l 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 + k-l K lc«l ^ P-647 .6-9- 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 &, ~ (1-w) Z k-1 K kH-r++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 - (1-w) t E ) 1 k*l k*t-r. +1 t t _ t represents the probability of condemning more than S ¦«• £ a,. - (1-w) E f k-1 k k-t-rt+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 [l-F (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 pile-UD of reparable items undergoing repair. The terra b represents the s o P-647 -70- 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: P-647 ^ + 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 sub-optimize 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 (x-S")dP (x) .0 4j .1 -" si is7 s" J 4J r, b ^ H f6 (x-S")(x+l-S") 4j s J-? J T^T) In equation (46) the first four terms are similar to the same terms in P-647 -75- 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(S-S ) - c] (1-w) Z f4k + f (o-w) dai(w) L k-ti-r, t-»-l ^o J t» where 0" H +" -i-at1- t c, ^ s s k-tt~rtt+l P-647 k -76- 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 life-of-type 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. t-pt 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 k-1 ^ 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 open-contract 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 sub-optimize 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 sub-optimal 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 sub-optimize procurement under constraint of the fund limit. a»-.—i-i-i IM an—a mummi P-647 -77- For this purpose, we may add the terra riU(S)S to all life-of-type 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 end-item 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 P-647 -78- 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 (x-S) (x-S)(x-<-l-S) dF0(x) s 2 >(x+l) Here, b (x-S) is the cost, which varies with the amount and duration 5 2 of depletion, of (x-S) 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 phased-out 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 phase-out; 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 .-79- 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 P-64'f -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 life-of-type 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 P-6/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 j-th 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 j-th 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 j-th major period. (8) FD.(x) « ( fn,(t)dt ¦ the cumulative demand probability function over minor period p in the j-th major period. (9) ÖD4 "' " the average cycle between receipts from repair in the j-th 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 depot-to-base 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 T-Sltf 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(x-0dx 00 (x~S ,+A r n.+w) . v / .] gj M.(SD.,x»S -z^.-Wjw) dF(x)i-da)(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