Stock management after catastrophe the economics of hold or destroy

Stock management after catastrophe The economics Of hold or destroy John B. Westwood, A. Geoffrey Lockett and K. Pennycuick

tmphe" which is unaveidable. This paper suggests a simple model for the control of stock after such an o¢.c~ttxeflce. A simplified cash flow model is developed as a special case of the standard dynamic programming formulation. This relates an initial scrapping policy to production for stock at a later date. Costs ate assumed to be time-dependent in order to examine the effects of inflation and discounting on the optimal scrap/hold policy.

Mandamer Business School, Booth Street Weu, Manehester MI $ 6PB, England

2. Operatingdmaetedstios

ReceivedMatch 1976

Demand

The l~oblem of stock con~ol ofter a dramatic deereasein demand is comidemd. This paper pze~ats cash flow modeh of hold or destroy which are appmpxiatein such a situation. O~rtain assumptions are made which enable a simple model to be produced which takes into account the effects of inflation and incorporatescash-flowdiscounting.The results from this ate then compazedwith some commonheuristicsand ate found to have lower cmts in every case. I. Introduction

The majority of stock-holding models assume either that demand is constant or that its variation over time is not too dramatic. Analysis of the demand-time function has usually been restricted to identifying seasonality or linear time-based trends; see

for example [2-4]. The work which follows looks at the particular problem of decaying demand for products when production has ceased. Typical examples of this occur in the motor, electronics and electrical appliance industries. If a product is discontinued, the "new' demand for many of its components will fall to zero, while the 'replacement" demand may continue for several 3"eara. In many instances technological innovation makes the planning of stock run-down extremely difficult. We may have a large on.hand stock ant greatly reduced demand, i.e. a management 'catas.

If the catastrophe occu@ at time to then we assume that demand will fall from some initial value /)(to) to zero some time later. A negative exponential function exhibits these properties and apart from its simple structure has a realistic decay profde. We can therefore write: D(t)=D(to)exp(--k(t-te))

t~te,k~0

(1)

where k is the decay constant. The overall demand is shown in Fig. 1 wt,ere the 'new' and "replacement' requirements have been separated. At time to, the new demand is zero and the replacement demand immediately starts to decay. According to eq, 1, demand does not fall ~o zero until t is infinite but this is an unrealistic bomdary condition with which to work. We therefor~ .~sume a finite life and impos~ a cut-off in demand. Suppose for example that the number of a popular m rice of car in existence is 1,000,000 and that production has just ceased. It is estimated that the last vehicles will be scrapped in 23 years time (i.e. t - to = 20 yrs) and that replacement demand for a particular component in all future years will be 25% of the existing population. Let us consider the life to terminate when demand drops to 1 unit per day. If we also assume 250 demand days per annum, on substituting these values into eq. 1 we obtain the expression: 1 = D(t o) exp(-5000k).

The authors would like to thank Mr. A. Stark of the Manchester Business School for his suggestionson some of the f'_manO.alaspects of the paper.

As the cut off demand rate will usually be small, time will be measured in days throughout the rest of the paper. If sales of 250,000 components are estimated for the next year, we can integrate the demand func-

~)North-HollandPubflshingCompany European Journal of OperationalReseatchl (1977) 94-102. 94

J.B. Westwood, A.G. Lockett, K. Pennycuick / Stock nmnagementafter~t~rophe

95

T New~

....................... - T - -

T °:)to

o

t4m~_ ._~

Fig. 1. Demand plofiAebefore and after catastrophe.

tion and write: 250,000 = D(t°) [1 k

-

exp(-250k)].

By inspection, the pair of equations are found to have an approximate solution of D(to) = 1200, k = 0~3014. In other words to establish the decay rate and initial demand, two estimates concerning future demand are required. In our example we have looked at the life of the component and its replacement demand in the first year after to.

Swck As we shall demonstrate at a later stage, the scrap/ hold decision depends critically on the amount of

stock which is on hand at to. This in turn is a function of the relative magnitudes of new and replacement demand. Consider the two cases shown in Fig. 2. The total demand immediately prior to t o is D and let the initial replacement demand at t o be D(te). If the decay rate is k, the total demand to extinction is approximately (found by integrating the demand function to infinity) D(to)/k and so the factor (l/k) is the number of days stock at initial demand. Further, let the ratio D(to)/D be p(0 < p < 1) and let the company policy prior to to be to hold on average q days stock. This means that immediately prior to to we have a stock level of D.q which is equivalent to (D.q)/D(to) = q/p days supply of initial replacement demand at to. In Fig. 2(a), p = 02.5 and let k = 0~)1 and q = 30 giving q/p = 120 and 1/k = 100. This

t

t Dmmd

ID(t~ to

(.)

tim

.-b

D(tO/U ...zs

Fig. 2. Initial stock-holdingpositions.

to 0)) D(to)]D = .75

t~

--~

¢J6

JJ. Westwood.A.O. Lodcett. K. Pennycuielc/ Stock nmnagementalter ~ p h e

means that there is a stock of 120 days at cunent demaml whereas the life to complete extinction is oily 100 days. At this point the obvious suggestion is to scrap 20 days demand and let the stock run down to zero. Ik, peading on the relative costs of holding and production, however, it might be cheaper to scrap more than the 20 days demand and produce a further batch at a later date. In Fig. 2(b), p-- 0.75 and with sinular values of k and q we have q l p = 4 0 and 0/k) -- 100. We have on hand less than half of what is required so the problem here is more the classical production type where we must decide wtten to produce and how much. Costs The model we develop treats the various ways of satisfying the demand during decay as capital projects requiring certain cash injections at particular points in time. We make the assumption that all demand will be suppfied on request (i.e. no bacHogging allowed) during decay, so the total revenue will be constant whatever production/stocking policy is enforced. For this reason we take the minimisation of total cash outflow as our objective. This emplmsis reflects the present concern over the huge amounts of cash tied up in finished goods and work in prcgress. It means, however, that departure must be made from the traditional view [8-10] of holding or carrying costs. These are usually assumed to cons;st of: the return which could have been earned on the money tied up in stocks, storage costs, taxes on stock, obsolescence, insurance, others (breakage, pilferage etc.). The opportunity cost is usually the largest and is generally represented as a percentage of the value of the stock at cost. Such a cost, however, is not a cash f o w and therefore has no place in our model. The remainder of the costs are proportional to the amount of stock on hand and represent cash fl~w out of the company. We can therefore use a figure of £/unit stock/unit time. The advantage of such an approach is that the resulting value of the objective function is an estimate of the actual cash needed to implement the policy, and the solution also includes an indication of when such cash flows will be required.

3. Problem formulation Given that the demand during the life to 'cut-off' is known, the question arises as to whether any exist-

in8 stock above this level should be retained. Clearly the only retention worth considering is an insurance against understating demand. H o w e v e r , this is o n l y necessary if the particular spare is to be regarded as a stock item for a specified number ofyears. Otherwise destruction (or salvage) of surplus stocks seems sensible. This is especially important if facilities exist for small-batch production. The relevant cost elements are then: (a) Cs - salvage value per item at time to, Co) CH(t) -- stock-holding costs per unit time per stock item, (c) Co(t) - production set-up costs at time t, (d) Cp(t) - production costs per item at time t. It is assumed that the time required both to scrap items at time t o and to produce items at time t is negh'gible. The problem we are considering is of a multi.stage nature and may be approached using dynamic programming [1,14,15], or the discrete maximum principle [5,11]. The latter may be used because the demand decay which we have described approximates to a continuous process. In many cases, however, the application of either method to production/storage problems yields solutions which require oniy one discrete change in the system. For example the composition of a vehicle fleet should be changed only once to satisfy future delivery requirements; a warehouse should be increased in size on only one occasion; a configuration of depots should be altered only once to handle demand over a specified number of periods. If such an assumption is made a priori, then the sophistication of dynamic programming or the dis. crete maximum principl ~. is unnecessary as a much simpler format is possible. For the problem under investigation therefore we will assume that at most only one furthe.r production run will be necessary at the time of, or after, the catastrophe. This does not necessarily mean that further production can never decrease total costs; the advantage of a simple model based on the assumption of one further production run must be compared with the possible non-optimal. ity of the solution. Assuming that we have estimated the decay rate (k), the initial replacement demand (D(to)) and the time at which demand effectively ceases (7"), we will develop models based on the following steps: (1) select a time tx(to < tt < T), (2) scrap anything not required between to and tl, (3) produce at tt enough to last from tl to 7".

J.B. Ifestwood, A.G. Lockett, K. Pennycuick [ Stock management after catastrophe

so

97

! I ! -

w

to

time

~

tl

T

Fig. 3. Stock profile in the generalcase. When no systematic analysis is performed, the production/storage policy is generally based on simple heuristics. For this reason the developed model is tested against two of the more common rules employed, namely: (i) if and when the present stock runs out, produce enough to last until extinction (i.e. no scrapping is involved), (ii) produce enough at to so that stock will last until extinction. Although rule (i) could be generated under certain cost conditions by the model as stated, rule (ii) could not. Setting tt -- to in the model effectively means that all stock is scrapped at to and simultaneously enough to satisfy the total demand is produced. As scrap value wiil inevitably be less than production cost, such a solution will never be generated and in some extreme cases we would expect rule (ii) to produce lower costs than the proposed model. To complete the model, therefore, we calculate the cost of producing at to and compare it with that of producing at the derived value of tt. This is operationally easier than using a more general and hence more complex model. Once the point tt in Fig. 3 has been established, we can calculate the amount to be scrapped and the size of the production batch. The initial stock is So and the demand will effectively cease at time T, having followed an expot~ential curve from time t~ Time.dependent costs (Model 1) As we are concerned with cash flows over an extended period of time, to be useful in a decision-

making situation, the model must ~adude the important elements of inflation and discounting. We extend the example of Van Home [13], and include cost inflation in the same form as the discount factor (see also [7]). Let the estimate of the inflation rate beg% per annum and the estimate of cost of capital be i% per annum. The various costs over time are the terms of a simple geometric progression and can be expressed in the form: CH(I +g/365) t-t° c.(t) =

(l + i/365) t-~°

Where CH is the value at to and t is measured in days. The numerator represents the cost increase due to inflation, while the denominator reduces future costs to a value at time to, when the hold/destroy decision is made. We can at this point set t e = 0 with no loss of generality and hence D(to) = D(0) etc. Let us write a H = (1 + g/365)/(1 + //365) and hence CH(t) = CHartt. If the rate of cost increase is different for production set-up and variable costs, then we have Cp(t) = Cpapt and Co(t) = Coaot where ap and ao are modified accordingly. We now turn to one of the most important requirements of cash flow models, which is that cash flows must only be related to the point in time at which they actually occur. As far as our model is concerned, there is no problem with the production costs as we assume that they are payable at the time of production. Holding costs, however, are made up of several elements which may be due at different intervals, e.g. weekly, monthly or annually. To eliminate the need to separate these costs, and also to make the

l~t.WeWwood,A.G. t~wkm. £. P m ~ y a ~ / Smet man~emem a[ter~ u o P he

98

eemputat~ suaplm, we mune~thatholding me a conenuous cash flow, i.e. ate payable daffy. This is ~ not the case, but R is e a J y s h o w n that the e,io~ is less than 1% when c~npared with monthly payment using reasonable estimates for inflationand the cost of ~eital. There are four distinct cost categories and these me described individually and then combined to produce a total cost function.

Reverme from salvage (C1) The stock needed (Dtl) to last over the period t = 0 to t = tt is given by: Dh=

f

D(0) D(t)dt=--~--[l_exp(-ktt)].

(2)

o

Anything in excess of this can be scrapped, and if $o is the stock on hand at tirre t = 0, the negative cost of salvage will be:

I'D(O) So]. C1 = CsL--~-~ [1 - exp(-kh)l -

(3)

Stock.hoMing cost from t = 0 to t= tl (C2] The stock level $(t') at any time t' must be such that at t = tt it has fallen to zero, i.e.

dC2 _ C x D ( O ) e x p ( - ~ t ) ( a ~ dtt

D(t)dt',

- l)

(5)

loge all

This is no longer a component of the cost function, but a component of the total derivative. Production cost at t z(C3) The production level will be such as to supply demand from time t = tt to t = T, and so the cost wmbe

T

Ca=Coat~ +Cpa~s f D(t)dt tl

(6) ffi Coa~ ÷ r a q D(O) [exp(-ktt) - exp(-kT)]. '~P P

k

Stock-hoMing cost from t = t l to t = T In this case the stock level at time t' will be given by: T

$(t') = / D(t)dt ,

tl

$(t') = /

where u t and u o ate differentia" b ~ functio~ Of a, and f . is the partial derivative off(-.t, a ) with ~ t 0 ee. we can now write:

tt ~ t' ~ T,

Ogt'~gh. and the cost of holding stock over this period will be:

The cost of holding stock over this period is therefore given by:

C2= /

tl

' tl

CHaHS(t ¢ , )dr_

/

T

t° Cxax[exp(-kT) - exp(-kt')]dt' .

rl

ti

f Cxa't

0

D(t)dt]dt'.

Application of eq. 3 yields

de, = Clta ~ D(.~0)[exp(-kT)

This reduces to:

dr1

C2 = --CH f i a ~ - ~ [ e x p ( - k h ) - e x p ( - k t ' ) ] d t ' . o Given the complexity of this expression it is simpler to differentiate under the integral sign by making use of the following expression d

c, =f

ut(a)

du l

-~a uof(a) f (x, a) dx = f(ul, a) - - ~ ..i'i't,'i

exp(-ktl)].

(7)

If we now differentiate eq. 3 and 6 and add them to eq. 5 and 7 we obtain the 'derivative of the total cost function. After some manipulation (given in the Appendix) we arrive at the following condition for optimality: exp(_kh)[- r * Cn(a ~ - 1) L~, logeaH /'w

..1'!1~

~

I"L~__.,w~l"l

LB. Westwood,A.G. l~ok~t, K. Pennyeu~k/ Stock managementafter catastrophe

99

4. Model results =

l"

A numerical example

Ce~dlogeao

(8) Although it is extremely difficult to solve this expression analytically for tt, a simple "trial and error' approach enables the solution to be derived very qmckly. We now have to retum to eq. 2 to test the feasibility of tt. Dtt is calculated and two possible situations arise, namely: (a) So - Dti ~ 0 - in this case the optimum value is that derived from eq. 8. (b) $o - D q < 0 - here there is not enough stock at time t = 0 to last until the time t = ti produced by eq. 8. In this case tt must be selected so that: tt

So = f D(t)dt, o

S o = 12,000 units

D(0) = 10 units/day

k = 0.007/day

T= 3,300 days

Cp = £10/unit

G = ~6/unit

CH = 2.5/unit/annum

i = 25%/annum

= £0.007/unit/day g = 15%/annum.

co = zl,OOO We assume that the inflation and discount rates apply to all costs, so we can write: an = ao = ap - (1 + 0.25/365)/(1 + 0.15/365) = 1.0003.

i.e. So = D!O) [1 - exp(-ktt)]. g

(9)

To find the size of the production batch (P), we sire. ply calculate DT, the total demand over the life product, i.e. T

DT = f D(t)dt o and P is then given by

P=DT-Dtt

S o - D : t ~>0,

P=DT-So

$o-Dt~ <0.

(10)

Time independent costs (Model 2) If we simplify the model even further and ignore any time.dependence of costs then the condition for optimality can readily be modified by letting arj, ap, ao tend to unity. Making use of the expression

a-~tLIogea.! we can reduce the condition for optimality in eq. (8) to:

F

Lee us consider an example with the following system data:

e x p [ k ( T - ti)l I. I I - kti

k(c.-c,) l_ ~ ] - 1(11)

Solution of eq. 8 gives tt = 1.947, and D1947 is then found from eq. 2 to be 10,630 units; as So - D1947 > 0, tt = 1947 is the optimal solution. D33oo , the total demand, is found to be 12,850, and using eq. 10, the production batch size is 2,200 items. The optimal two.stage policy is therefore: (1) scrap 1,370 items at time t= 0 (2) produce 2,220 items at time t = 1947, as is shown in Fig. 4. To fred the corresponding result from Model 2, we write kq = x in eq. 11, and obtain the following expression: exp(-x) (1.4 - x) = exp(-2.3). This can be solved by making the transformation Z = 1.4 - x and plotting the function f(Z) = Z.exp(Z) for various values of Z. The optimal policy is then found to be: (1) scrap 2,470 items at time t = 0 (2) produce 3,320 items at time t = 1571, and this is seen to differ appreciably from that produced by Model 1. To test the two models, the sample dam already specified was used. For each run one of the parameters was varied and all others kept constant. The results are ~ o w n below in Table 1. The major point of interest is that the value of tt produced by the simpler time-independent model is in every case considerably less than that produced by Model 1 (we shall refer to results from this model as "true' values). For Cp = 17 and 20 the true values of

|08

J.B. Wejriwoad,A.G. ~ ,

£. Pesnycuick/ Stock mnnqeme~ after catastrophe

IA~el I

_

0

jz2o_

_

1947

time

$~,00

Fig. 4. Stock pzofile for numerical example. costS produced by Model 2 are on average around 33% too l¢~w and the values of tt are about 20% too low. The true cost associated with such values of tt are, however, only about 3% higher than optimum. Model 1 d~ows the effect of inflation and cost of capim~ unc]er different economic conditions to be investigated. It would seem, therefore, superior to Mode! 2, esp,~cially as there is virtually no difference incomputational costs.

tt are in fact greater than T so the operational values of tt will in fact be determined by eq. 9. The simple model, however, continues to produce feasible values through eq. 11, the condition for optimality. Table 2 gives a more direct comparison of the two models. The tint two columns show the inability of Model 2 to produce either tl or a cost estimate in the region of the true value. Column 3 shows the difference between the true cost associated with the optimum tt and the true cost (using the Model 1 cost function) associated with the Model 2 value o f tt. In summary, for the example investigated, the

Comparison o f Model 1 with simple heuristics T~3 test ~ e possible uses of such a model, the rew~ts already produced were tested against the

Table 1 Computational results for comparison of the models Parameter change

Model I tl

Batch size

True cost

tl

B~.tch size

Cost

True cost b

Examplea CH = 0.01 CH=~004 CO= 10000 C0=500 Cs = 9 Cs = 3 Cp= 17

1947 1721 2618 ¢ 1736 1961 1506 2292 2618 d 2618 d

2220 2850 850 2810 2180 3540 1440 850 850

127029 164761 85594 142643 126131 121128 129870 146991 153916

1571 1429 1900 1571 1571 1214 1860 2285 2571

3320 3850 2280 3320 3320 4660 2450 1420 920

82987 109265 55484 91987 82487 74255 88516 99388 103130

128583 166376 88804 142998 127782 122091 131906 151027 154766

Cp=20

Model 2

a The basic data is that used in the example for Models I and 2. The parameters were changed one at a time. b This is the cost of using the value of t I generated by Model 2, but calculated using the cost function of Model 1. c The calculated value of rI is 2898, but the demandin that period of time is >12,000, so in this case t I must be determined from the the valu~ of So. d The c~lculated value of tl is >3300, so again S 0 is used.

J.K Westwood, A.G. Lockett, K. Pennycuick / Stock management after catastrophe Table 2 Comparison of remits of the two models

% Difference

Paramete~ change

~ Difference between values o f t t (tn~ Model 2)time

% Cost diffesetice (true Model 2)/tn:e

between true colts a

Example CH = 0.01 CH -- 0.004 C0ffi 10,000 CO500 Cs = 9 Cs = 3 Cp = 17

19.3 17.0 27.4 9.5 19.9 19.4 18.8 17.3

34.7 33.7 35.2 35.5 34.6 38.7 31.8 32.4

-1.2 -1.0 -3.8 -0.2 -1.3 -0.8 -1.6 -2.7

7.0

33~

-0.6

cp ffi20

aThis is the true cost for the Model 1 solution (Table 1, column 4) minus the tree cost of the Model 2 solution (Table 1, column 8) divided by the true cost for the Model 1 solution.

following heuristics: (1) If and when the present stock runs out, produee enough to last until extinction. (2) Produce enough at time t - 0 so that stock will last until extinction. In our examples, So is 12900 and the total demand is 12,850 so we do not have enough stock on hand to last until extinction. The results are shown below in Table 3. As explained previously, policy 1 can be produced by the model, but policy 2 corresponds to the evalua. tion of the 'end point', i.e. the costing of production Table 3 Comparison of simple policies with Model I Parameter

Policy I a

Policy 2 b

change

tl ffi2618

tl = 0

Example C~I = 0.91 CH ffi0.004 CO= 10,000 C0--500

Cs=9 Cs = 3 Cp = 17 Cp = 20

Cost

% difference Cost from optimum c

130834 176074 85594 150564 129738 130834 130834 146991 153916

3.0 6.9 0 5.6 2.9 8.0 0.7 0 0

%difference from optimum c

144371 13.7 202173 22.7 86569 1.1 153371 7.5 143871 14.1 144371 19.2 144371 11.2 150321 2.3 152871 -0.7

a Produce 850 when present stock of 12,000 runs out. b Produce 850 at time t 0.

¢ 'Optimum' is the solution given by Model 1.

101

at time t = 0. We find the optimum solution by comparing the costs of rite two cases i.e. production at t = 11 and production at t = to. It can be seen that in only one case (Cp = 20) was the end point in fact the optimum. It can be seen that policy 1 produces results which on average are much closer to the optimum than those produced by policy 2. No generalisa. tion should be drawn from tiffs, however, as it is related to the cost structure of the system. For exam. pie, if production costs are rising rapidly we would wish to produce as early as possible and policy 2 would be cheaper. Most practical cases are not as well defmed and the model developed in this paper allows the optimum decision to be taken whatever the cost structure of the system. 5. Discussion The major assumptions of the model are: 0) only one further production run is necessary, (ii) production time at t~ is zero, ('fii) stock-holding costs are payable daffy, (iv) k and D(0) can be estimated. The first assumption is necessary in order to obtain the simplified model from the general multi. stage format; if it were relaxed, the computational procedure would be much more complex. As a test of this assumption, the examples investigated were expressed in both dynamic programming and discrete maximum principle formats. It was established that in neither case did a second production run result in lower costs than those of the optimal strate. gies generated by the simpler model. The second assumption can obviously never be perfectly satisfied, but the production time for a small.batch run may well be very short relative to the accuracy required of the model. Assumption Cfii) is another that can never be satisfied in the practical sense but the error involved is small and the model is greatly simplified. We have explained earlier how the parameters and D(0) can be calculated from: (a) an est~nate of the component's life and (b) the relative value of new and replacement demand. These two quantities are both subject to error, the size of which will depend on the type of component and the particular industry and no generalisations can be made. The ease with which the model produces results, however, enables considerable sensitivity analyses to be camed out. Our approach has been similar in many ways to the replacement problem which has already received

7~t w v ~

102

~Lo. ~ . e u , g. Pmnyaect / sto~ m~eem~t ¢e~, an~moeke

mmiderable attmeon [6,t2] pmicutady m the area of discounted Crab flows. For out example we have ~hewa that the polly phi'rated bY ~ t ~ ' l i e supeto other common heuristics. Little time is to find the optimmn solution and the importam factors of inflation and discounting ate easily incorporated to give an e~mate of the cash-flow

profile rather than the traditional accounting cost. enables stock holding to be compared on a common basis with alternative uses of funds which may be open to the company.

-t

CSaftO(0) [exp(-kT) - exp(-~t)] = O. k

i ' e ' e x p ( - - k t l ) [ CsD(0)

CHa~D(0)k

Cpa~'D(O)

apD(O) + +

k

~=D(O)exp(-k'T) [Cpa~llogeaP k

J CHaff] k

- Coa~Hogeao.

Appendix

Dividing through by D(0) we fmally obtain eq. 8.

To obtain the condition for optimaliW we set the total first derivative equal to zero, i~.

Refmees

dCl +dC2 + dC3 + dC4= 0" dtt

dtt

dtt

dtt

From eq. 3 we can write:

de, = c o ( o ) dt~

k

[k • exp(-/~t)]

(A.1)

= C.D(O) exp(-ktt). From eq. 6 we can write:

dC3 dt t

D(O)e - k T = Coa~ Ioge ao - Cpa~11ogeap

k

Cpa[l D(O) k • exp(-ktt)

k Cp ap loge apD(O) exp(- ktl)

(~L2)

Combining eqs. 5 and 7 with (A. 1) and (A.2) we have CsD(0) exp(-k71) + Co a~tloged0 - Cpa~t loge apD,(~ O) exp(-kT) - Cpa~'D(0)exp(-kt,) +

Cpa~1 logeapD(O)exp(-ktl) k

~_CHD(0) exp(- kt l ) (a~ - I)

[1] g. Bellman, Dynamie Programming (Princeton, 1957). [2] G.F. Brown, TaM.Corenran and K~M.Hoyd, lnventow models with forecasting and dependent demand, ManagementSci., 17 (7) (Match, 1971). [31 S~i. Chang and D~. Fyffe, Estimation of forecast errors fog seasonal-style-goods sales, Management Sci. 18 (2) (October, 1971). [4] W.B. Cmwston, W.H. Hausman and WX. Kampe, Multi-stage production for stochastic seasonal demand, ManagementSol. 19 (8) (April, 1973). [5] L.-T. Fan and C~. Wang, The Discrete Maximum Principle: A Study of Multistage Systems Optimization, C#aey,NT, 1964). [6] B. Fox, Age replacement with discounting, Operations Res., 14 (3) (1966). 17] K.P. Gee, Capital project appraiss] in inflation: A survey, unpublished paper presented at the British Accountants and Finance Association Conference, Manchester Business School (1974). [8] G. Hadley and T.M. Whitin, Analysis of Iuventory Systems (Prentice-Hall,NJ, 1963). [9] BJ. talomie and DaM. Lambert, Inventory carrying costs: sigltificance, components, means, functions, lntemat. J. Phys. Distfib. 6 (1) (1975). [10] E. Naddor, Inventory Systems OVHey,NY, 1966). [11] L,S. Pont~yagin et al., in: L.W. Neustadt, The Mathematics! Thcow of Optimal Processes (Wiley, NYj 1962). [12] B.D. Siva~ian, On a discounted replacement problem with arbitrary ~ time distribution, Management Sci. 19 (11) (July, 1973). [13] J,C. Van Home, Financial Management and Poficy (Prentice-Ha]i,NJ, 1968) Chap. 5. [14] HaM.Wagner and TaM. Whitin, Dynamic version of the economic lot size model, Management SCi. 5 (1) (October, 1958)., [15] J.F. wmlams, Multi-echelun produotiun scheduling when demand is stochastic, Management SCi. 20 (9) OMav.1974~.