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Ordering and stockholding under price inflation when prices increase in successive discrete jumps
EJzg~~~e~jngCosts and ~redMct~en Economics,
395
I 7 ( 1989) 395-407
Elsevier Science Publishers B.V., Amsterdam -Printed
in The Netherlands
ORDERING AND STOCKHOLDING UNDER PRICE ~N~~AT~ON WHEN PRICES INCREASE IN SUCCESSIVE DISCRETE JUMPS* Brian G. Kingsman and Aziz Boussofiane Manufacturing Systems Research Group, Department of Operational Research and Operations Management, The Management School, University of Lancaster, Bailrigg, Lancaster LA 1 4 YX (Great Britain)
ABSTRACT
.l/last purchased materials and supplies are subject to increases in price over time. Thus a vitul asszlmption qf classical inventory theory, t~~)?~.~ta~t unit prices, is incorrect. Prev~o~~~~ work in this area has ass~~~~~ed either ~ontin~~o~~s esponentialgrowth in prices or that the data qfthe ne_u price rise is known in advance. Investigation showy thut pricesproceea’ via a series of discrete jumps ,~~)~~~)~,ed by plateaus. The time be-
tween successive jumps is stochastic. This paper makes use of‘some analogies with renewal theory to produce appropriate ordering and stockhording poIicies. The model and some interim results are described. There are two critical Ievels for inflation, one below which inflation plays no role in the determination of the order quantities, and a second above which a single purchase stralegj? is best.
1. INTRODUCTION
ilar inflationary situation, to a greater or lesser extent, occurred in all developed economies over this time. Because of this, suppliers became reluctant to guarantee prices for much longer than three months. A basic assumption of classical ordering and stock control theory, constant prices for the indefinite future, was thus broken. Yet, the effect of price inflation on purchasing and stockholding policies has apparently received very little attention in the published literature. Governments changed policies to attempt to bring inflation under control. In the U.K., inflation in industrially purchased materials fell to zero between 1977 and 1978, then rose to 10% per year and remained around this level until 1984, see Fig. 1. Although prices fell since then, there is a continuing possibility of high
Prior to 1970, price inflation in developed economies was small. typically around 4% per annum or lower, so buyers could rely on the prices being offered for component parts, manufactured and semi-manufactured materiais being stable for periods of a year ahead and longer. The increases in price were usually quite small. From 1973 to 1977, inflation for industrially purchased materials in the U.K. was running at an annual rate of 25-300/o, see Fig. 1, which shows the annual values of the U.K. Index for Materials and Fuel Purchased by British Industry from 1966 to 1987. A sim*Presented at the 5th International Working Seminar on Production Economics, I&, Austria, February 22-26, 1988.
0167-188x/89/$03.50
0 1989 Elsevier Science Publishers B.V.
396
300
200
100 tJ 0’ 64
” 66
Iear ‘1
68 70
”
72
74
‘1’
76 78
8002
Fig. 1. Index of prices of U.K. industrially rials 1964-1987 (base of 100 in 1970).
0
’
84
86
purchased
mate-
levels of inflation reoccurring. In developing countries high inflation is apparently endemic and has been so for many years. Inflation rates of over 100% per annum are not uncommon occurrences. So exploring the effect of price inflation on ordering and stock control theory remains an important practical problem. Buzacott [ 1 ] extended the classical EOQ to cover this situation, assuming the same exponential growth in all the inventory cost components as in the cost of the purchased material. To a very close approximation, he showed that the optimal purchasing and ordering policy is the same as the classical EOQ but with a modified inventory fraction holding charge, I*, which depends on the final product pricing policy used by the manufacturing company buying the material. If the final product price is independent of the purchased materials’ cost, then I* is the usual value minus the annual inflation rate. However, if the inflation rate is larger than the cost of holding stock, as it was for the U.K. in 1973-1977 and in the U.S. for some years, the model has no solution. All that can be derived is one initial purchase of an in-
finite or indefmite quantity of material. If the final product price is a fractional mark-up on total costs, ( 1 + k), then I* is the normal value plus k times the inflation rate. This gives more frequent purchases of smaller quantities than the classical EOQ. The motive behind this is to adjust one’s prices rapidly and frequently to reflect inflation. This would imply that the profits of industry should generally have maintained their value over time. The fact that this was not true for British industry, for example, over these years, is probably an indication that this is not a final product pricing probably in general use. Finally. if the final product price is a constant mark up on the materials cost, as occurs in some sectors of the non ferrous metals industry, then 1* is the normal inventory holding fraction charge. The same assumption of how inflation occurs was used by George [2] and Jagieta and Michenzi [ 31. Their results were very similar to those of Buzacott [ 11. The assumption of a continuous exponential growth in the unit price over time appears reasonable on a price index basis at times, see Fig. 2 which shows the same index as Fig. 1 on a monthly basis for the high inflation period, 1972 to mid 1977. As can be seen, the index values fit very closely an exponential growth curve lndex=92.76exp(O.O213T)
where T= 1 in January 1972, 2 in February 1972, etc. The temporary surge above the growth curve in late 1973 was the effect of the fourfold increase in oil prices following the Yom Kippur Arab Israeli war. Whilst possibly true for an index of materials, the assumption of a continuous exponential growth in prices is not valid for individual materials. Suppliers prefer to provide more stability in their prices by making step changes in the price at infrequent intervals of time apart. Some examples of the actual prices charged by suppliers over the high inflation period are given in Fig. 3, for four products
397
Fig. 2. U.K. Index of materials and fuel purchased by manufacturing
7rJrJ _
100
industry.
Unit Price in f per tonne
-
0
1
1973
I
1974
I
1975
1977
Fig. 3. Examples of inflation on prices of individual materials.
bought by an industrial company for use in producing a final product. In late 1973 and early 1974, when high inflation was a new and strange phenomenon, there were frequent small price increases in some cases. Thereafter, suppliers introduced more stability into their pricing. Thus as well as not giving a solution in many circumstances, Buzacott’s model is based on a false picture of how inflation affects the prices of individual materials. Often the step
increases in price were similar over time for a particular product. What seemed to change was the time between successive increases. An alternative approach has been taken by Naddor [ 41 which partially fits the circumstances of step changes in price. He considered what quantity to buy just prior to a known single step price increase, known both in size and timing. Let c and c* be the old and new prices, D the mean annual demand, I the inventory
398 holding fraction charge and Q and Q* the Wilson EOQ values for the old and new prices. The solution is to continue to buy in lots of size Q until the day prior to the price increases. On that day a purchase should be made of an amount
Once this quantity is used up, the material should be purchased in lots of size Q*. The Naddor model was largely an academic exercise. It has been further developed by amongst others Lev and Soyster [ 5 1. Their emphasis was on determining better what orders to place from the date the price rise is announced to the date the rise actually occurred. However, the above model still plays the central role in calculating the size of purchase to make just prior to the price rise. Gee [ 61 also considered this model from an accountant’s viewpoint, on its implications for cost control procedures and the measurement of purchasing and stockholding performance. If there is one price rise per year and the inflation rate equals the inventory holding fraction charge, then (c*-c)/ (Zc) is equal to 1. Thus the recommended order quantity is the annual demand, D, plus Q* (c*/c). This quantity covers the demand over the period to beyond the second price rise. Hence the quantity should be calculated to cover two price rises, which by the same analysis will give a quantity covering demand up to beyond the third price rice, etc. The continuation of this analysis shows that when the annual total price rises are larger than the annual inventory holding rate, or more accurately a value just under the inventory holding rate, the amount to purchase just prior to the first price rise is an indefinite infinite quantity. Thus just like the Buzacott Naddor’s model fails in these model, circumstances. More importantly, Naddor’s model explicitly assumes that the date and size of the price rise are known well in advance. Advance notice of a price rise is sometimes given, possibly
up to a month in advance. An unstated but vital assumption is that the supplier will allow such a very large purchase, covering several months’ usage requirements, at the old price just prior to the day of the increase followed by a long period of no orders. It is rather unlikely that the supplier would accept such an order as it is completely contrary to his intentions in increasing his prices. If there are many suppliers and the consuming company randomly varies its purchases between them as a normal policy, it may be feasible. In the more general case of only one or two suppliers, they will have a pretty good picture of their customer’s usage pattern and so know the buyer is trying to take undue advantage of their advance warning of a price increase. Even if a larger order than normal is accepted by the supplier, he is likely in the future to react to the disadvantage of the buyer by increasing his prices earlier and by larger amounts than he would normally have intended. Clearly an ordering model is needed that anticipates possible price rises in advance. 2. A RENEWALTHEORY PRICE INFLATION
APPROACH
TO
From the examples of Fig. 3, the most realistic model of how price inflation occurs is as a series of step jumps followed by different plateau levels. It seems reasonable from the examples to assume that the size of the step increase is roughly constant, either in absolute or percentage terms. However, the occurrence and timing of the price increases are unknown. It is assumed that the time between successive price increases follows some known probability distribution, calculated from past data. To determine the best future ordering policy, requires estimates of the probability of having a certain price at any time in the future, knowing the past history of price changes. More specifically, since prices increase in a series of discrete jumps, it is necessary to know whether the price has increased since the pre-
399 vious purchasing opportunity and by what amount. The expected price at time t will be the base price at time zero plus the expected number of price jumps that have occurred in the interval [ O,t]. This requires the calculation of the distribution of the number of arrivals (price jumps) in the time interval [ O,t], where the number of arrivals will be integer, and the probability of having a price P, say, at time t+ At knowing that there have already been n price jumps by time t. The emphasis is on the number of price jumps and in the time between jumps. This requires a renewal theory approach rather than a queuing theory approach. We can draw the analogy between price jumps and machine breakdowns and between the time between two successive price jumps and the service time or life time of a machine. The probability of having a price P at time t+ At knowing the price was P at time t is equivalent to calculating the probability of having a machine in service at time t+ At knowing it was already in service at time t. This analogy means that we can directly use some of the standard results in renewal theory, e.g. Neuts [7]. Two probability distributions have been used to model the time between price jumps. The first simple case is a negative exponential distribution. Observations on a very limited data sample suggested that this did lit some cases. Furthermore, the calculation of the prices expected at particular future times is enormously simplified because of the ‘memoryless’ property of this distribution. This means that the probability of any price change does not depend upon what happened before, so all that needs to be considered is the probability of no arrival during an interval of time t. This is the one used to derive the results given in the final section of this paper. The second distribution is the Erlang distribution. This is a very general distribution enabling several different models of the inflation process to be evaluated. In particular, the probability of the next price jump occurring
can be made dependent on the time since the last jump occurred. It seems intuitively likely in practical situations that the probability of a price jump taking place will increase as the time elapsed from the last jump increases. Since the density function is a Gamma distribution, the calculations of the expected prices are again considerably simplified because of the standard well established properties of Gamma distributions. This analysis is continuing and will be described in later papers. 3. THE PURCHASING AND STOCKHOLDING MODEL The ordering and stockholding model is formulated for the case of a general inter-arrival time distribution. The objective is to minimise the costs of purchasing the known requirements over some planning horizon. The planning horizon is divided into N equal time intervals, e.g. months or weeks, such that there cannot be more than one price increase per interval. The demand for the item purchased is assumed to occur at a constant rate D per period. All demand must be met on time, so no shortages or backorders are allowed. A decision on whether to make a purchase or not is made at the beginning of each time interval. The problem is clearly a sequential decision making situation which can naturally be formulated in dynamic programming form. The decision at future time periods will depend upon the price prevailing then plus the amount of material in stock. Let fx(S,p) be the expected minimum costs of meeting the requirements over the remaining k periods of the planning horizon with a current stock of S units of material and a current price of p per unit. Let Po( k,n), where n < N- k, be the probability of having no increase in price in period k when there have already been y1price increases before period k. The price will then remain at p for the start of the next period k- 1. The probability of a price increase occurring is 1 -PO (k,n), whence the price at the start of the next period k- 1 will be
400 p+)??, where m is the jump in price. If a purchase is made to raise the stock on hand to Y,, then the resulting costs will be r;(S,p,~~‘,)=Sc(S,Y,)fpuR(S,~.y,) +INV(.~.a.Y~)+p,,(k,n).~~,(Y~-D,p)
+(l-P,,(k,n)If;~,(Y~-D,P+m)
(1)
where SC, PUR and INV are the set-up, purchasing and inventory holding costs incurred in period k. Then .~(S,P)=
Min
1, .I>.,; .\
.f,(S,p,Y,)
(2)
The set-up cost is an internal cost to the inventory/purchasing system. It will not normally be subject to the same inflation as in the price of the materials bought, particularly if these are imported. For simplicity, it is assumed to remain constant at a value A. Hence SC(S,Y,)=,-l =0
if Y,>.S otherwise
There is, however, no conceptual difficulty for the model in allowing the set-up cost to vary over time if there is internal inflation in the system. The purchasing cost is merely the amount purchased times the current price at the start of period k. Since the unit cost is p PUR(S,~.Y,~)=~(J,~-S) =o
if Yk;>.S otherwise, i.e. no purchase
made
The standard inventory holding cost model is that the cost incurred by a stock quantity, x say. is the average level of stock over the period until the s is used up in meeting demand times the cost of holding one unit of stock per unit time. This latter factor is taken as some inventory holding fraction charge times the unit cost of the item. Usually in dynamic programming inventory models, it is assumed that inventory holding costs payments are made regularly each period on the average amount of
stock hold over the period or the end of period stock. This is costed on the prevailing unit price of the item at the start of the period. Prices increase over time under the stochastic process specified. Although the value of the stocks held increases after an increase in price, the actual amount of money required to finance the stock remains the same. The same effort is required to look after the stock. The only factor that changes is the insurance component in the inventory holding charge. This is small and in most companies is only reviewed and changed once or twice per year. Hence the inventory holding cost should be calculated on the basis of the price paid for each unit of stock. This would require identifying each component of the opening stock of a period according to the price paid, thus requiring many state variables for the units of stock bought at the differing prices. This would make the DP (dynamic programming) model very complex and difficult to update. An alternative approach is to assume that the whole of the inventory holding cost arising from a purchase in period k is incurred in period k. This does not mean that the actual inventory charges are paid out instantly in period k. We are simply including the total inventory expenses that will be incurred in period k and future periods in period k rather than spreading them out over periods k to the time that order is used up, as is the usual approach. If, at time t, the stock in hand is s and a purchase is made at price p to raise the stock to y, this purchase will be used to meet demand over the period t + s/D to t + y/D, where D is the demand per period, see Fig. 4. The inventory holding costs incurred by the stock s have already been included in the model at earlier periods. Hence the total inventory holding cost incurred by the purchase y--s will be given by area (2) of Fig. 4, (y’-s2)/2D, times Ip. Hence, the inventory holding costs for period k to use in the model are INV(S,p,Y,)
=Ip/2D(
Y, ‘-4’)
(3)
401 The mean value 1/Y will represent the average time between price jumps, hence Yis the average arrival rate, number of price jumps per unit time. If X is the random variable representing the (continuous) time until the next price jump, then
Stock
Prob{X>t}=
1 -F(t)=exp(
-rt)
The ‘memoryless’ property is the so-called Markov property that for every t> 0 and x> 0 Prob{Xzt+x(X>t}=Prob{X>x}=exp(-rx) Time Fig. 4. Inventory costs.
An implicit assumption in this model is that stock is used up on a FIFO basis. The later stock may be bought at a higher price thus incurring higher inventory charges. It would thus be cheaper to use up the higher priced stock first. This implies that the inventory system is being managed to make speculative gains from the stocks held. This goes against the common practice of most companies. The gains from such a policy are likely to be small. It would make the DP model extremely complex as it would have to include all the possible delivery alternatives. An increased number of state variables would be required to keep track of the stocks bought at different prices. Since the gains are small, as mentioned, and the modelling difficulties large, this aspect is ignored and a FIFO use of stock therefore assumed. 4. NEGATIVE EXPONENTIAL TIME DISTRIBUTIONS
ARRIVAL
If a variable x follows a negative exponential distribution with mean 1/Y and variance 1 /r2, then its density function is given by f(x)=rexp(-rx)
forx>O
=o
for x-z0
The cumulative
distribution
F(x)=Prob{X
forx20
=o
forx
function
of X is
(4)
This is easily shown to be true for this negative exponential distribution. The ordering and stockholding model requires the value of PO(k,n ), the probability that no price jump takes place in period k, given that n price jumps have occurred previously. The ‘memoryless’ property means that this probability is independent of the previous number of price jumps, ~1,and the particular point in time reached, i.e. the period k. It is equal to the probability that there is no jump in price over the length of time in the period k. Since time has been divided into equal time intervals, the length of these epochs can be taken as the basic unit time. Hence putting x= 1 in (4) Po(k,n)=Po =Prob{X>
1)
(5)
= exp( -r)
5. SOLUTION
TO THE DP MODEL
The above value for PO can be substituted into the DP model. Unfortunately, it is not possible to determine a simple analytical solution to the model. It is necessary to solve the model numerically for particular sets of values for the various parameters. The solution to the DP model gives the optimal solutions at the start of each period as a function of the stock on hand at that time and the prevailing price. As might be expected the optimal policies are rather complex. A number of cases have been analysed and will be discussed in Boussofiane
[81. The major
problem
with all dynamic
pro-
402
gramming models is the large amount of computational effort required to find these solutions, even for quite small planning horizons. This can be reduced by increasing the length of the time epoch between successive price jumps. This makes sense for practical application since the time between jumps is likely to be months rather than days. Thus allowing one price jump per month is reasonable. However, the general model of Section 3, eqns. ( 1) and (2 ), and the derivation of PO, (5 ), assumed one ordering opportunity as well as one price jump per time epoch. Ordering opportunities more frequently than once per month are more appropriate in practical situations. This can be achieved by allowing L ordering opportunities between the times at which price jumps can occur. The basic unit time period is the time between ordering opportunities. Hence the time x in eqn. (4) now has a value of L time units, so the probability a price jump occurs is now P,,(k,n)=Pq
=Prob{X>Lj
(‘5)
=exp( -Lr) where l/r is the expected number of periods between successive price increases. If the arrival of price jumps is constrained to be only once per month at most, and there are L ordering opportunities per month, then the expected number of price jumps per year will be 12 Lr. The expected price increase over the year will thus be 12 Lrm. 6. SOME
GENERAL
RESULTS
However, it is possible to derive some general properties of the solution policy for the case of a negative exponential distribution for the time between price jumps. 6.1 The critical
inflation
rates
In the situation analysed, time is divided into discrete periods with an ordering opportunity at the beginning of each period. Since no short-
ages are allowed, the order quantity must cover at least the time until the next ordering opportunity. This time interval is taken as the basic unit time period. For simplicity at this stage of the analysis, all parameters are defined in terms of this unit time, so IC and D are the unit stockholding charge and demand over one period between successive ordering opportunities. It is assumed that if there is no inflation, i.e. the price jump m is zero, then the optimal order quantity, the rounded EOQ will cover the demand over one period. This will be the case if A < ICD
(7)
In this situation, the optimal order quantity will be to order 1 period’s demand at the start of each future period. Larger orders will only lead to increased stockholding costs. As inflation increases above zero, then there will come a point where it will be worth carrying stock to avoid a price increase in subsequent periods. In the general situation, no price increase can occur before the start of period L. Clearly, there is no advantage in buying other than a period’s demand at each of the first L - 1 ordering opportunities. The effect of inflation will be to increase the orders from the start of period L- 1 onwards. The initial impact of increasing inflation, therefore, affects the ordering decisions over the periods L- 1 and L, whether to have two separate equal orders or one order for the two periods. Let the arrival rate and size of the price jump where this occurs be denoted by r. and m,. The stock controller is indifferent between one order covering 2 periods’ demand and two orders each of 1 period’s demand, where the second of these orders might have to be bought at a higher price. Letting the base starting price be C per unit, the total costs incurred by an order of 20, covering 2 periods’ demand, placed at the start of period L- 1 will be 2 CD+A+O.S
IC(4 D)
(8)
403
The cost of an order of size D, covering only one period’s demand at time L- 1 will similarly be CD+A+Q.SlCD
If no price increase has occurred during the first month, then the second order of size D placed at the start of period L will have exactly the same cost. However, if a price increase has occurred, the unit cost of the items in the second order will be C+mo, hence the total costs of the second order will be (C+mo)D+A+0.5Z(C+mo)D
(9)
The probability of no price increase is exp ( - Lr, ), see eqn. (6 ), the probability of an increase 1 - exp ( - Lr,) . Multiplying expressions ( 8 ) and (9 ) by the probabilities of each event occurring, the expected cost of the two order policy simplifies to 2 CD+2 A+ZCD+{
1 -exp(
-Lr,,)}{rq,D+O.S
ImoD}
This will be the same as the cost of the one order covering 2 periods, expression (7) if DIC-A
mO[1-exp(-Lr0)l=D(0.51+1)
(10)
Note, that because of the condition given by eqn. (7) m, is positive or zero. As inflation increases then clearly it is worthwhile buying even more initially in period L- 1 at the base price. The precise effect on the order sizes depends upon the length of the planning horizon, the number of periods over which it is desired to minimise costs. However, there are many alternative possible sequences of orders so it is impossible to work out a simple solution. Analysis of a number of examples suggests that as inflation increases the number of orders placed to cover any given horizon decreases, but not in a simple way. For example, with a 12 period horizon there could be several alternative optimal ways of having only 4 orders, depending on the value of the price jump m. Finally, there must come a point when infla-
tion is so high that it is optimal to buy the whole remaining requirements for the planning horizon in period L- 1. Consider an N period planning horizon. The stage just prior to placing an order for (N-L + 1 )D, covering the remaining horizon, is to place an order covering the first N-L periods followed by1 order of size D covering the last period’s demand. The total costs for one order of size (N-L+l)Dwillbe C(N-L+
l)D+A+0.5
ZC(N-L+
1)2D
(11)
and similarly for an ordering covering N-L periods’ demands, the total costs are C(N-L)D+A+O.S
ZC(N-L)2D
(12)
The cost of the final order, D, at the start of the last period will depend upon the price at the start of the last period. Let P, be the probability the price is C+jm at the start of the last period. If the planning horizon is a multiple of L, the minimum interval between price jumps, then the maximum possible number of price increases over the planning horizon is (N / L- 1). If not, then the maximum number of possible price increases is the integer less than (N/L- 1). Hence for generality let K be the maximum number of price increases possible over the planning horizon. Thus the expected cost of the order size D at the start of the last period is ,to { (C+jm)D+A+0.5 CD+A+O.S
ICD+mD(
Z(C+jm)D}P,= 1+0.51)
F jP, ,=L
Since the maximum number of price jumps possible is K, the probability the price is C+jm is the probability there have been exactly j occasions in which an arrival has occurred and exactly K-j with no arrivals. The probability of j arrivals is [ 1 - exp ( - Lr) ]j and K-j non arrivals is [ exp ( - Lr) ]“-I. We are not inter-ested in the precise arrangement of arrivals and non arrivals merely the total number of arrivals. There are “C, permutations ofj arrivals
404
in the maximum
TABLE 1
possible K jumps.Therefore
Examples of the critical inflation rates for differing ordering opportunities per price epoch
Prob (price = C+ jm) = P, =KC,exp(
[ 1-exp(
-Lr(K-j)
-LT)]’ (a) Annual inventory holding fraction charge of 0.15
This is a binomial distribution with parameter exp( -Lr). The final term in eqn. ( 11) is just the mean value of this binomial distribution, which is K[ 1 - exp ( - Lr) 1. Thus the cost of the final order simplifies to CD+A+O.S
CD+mD(
1+0.5 I)K[ 1 -exp(
-Lr)]
Annual demand
0.5 IC(N-L+l)2D=A+0.5
IC(N-L)‘D+0.5
+m,D(1+0.5
ICD
I)K[l-exp(-LLr,)]
6.2 Examples
Fortnight (L=2)
IRo
IRo
KD(1+o,51)
of the critical
IR,
15.0 16.0 16.1
0.0 5.5 6.8
14.7 16.0 16.1
0.0 0.0 1.5
13.0 15.8 16.0
1,200
15 100 200
12.0 15.5 15.9
15.8 16.1 16.2
0.0 6.8 7.5
15.4 16.1 16.2
0.0 1.5 3.4
14.0 16.0 16.1
12,000
15 100 200
15.8 16.1 16.1
16.2 16.2 16.2
7.2 8.0 8.0
16.2 16.2 16.2
2.3 3.8 3.9
16.0 16.2 16.2
(b) Annual inventory holding fraction charge of 0.30 Annual demand
C
Ordering period Fortnight (L=2)
Week (L=4)
IR,
IR,
IRo
IR,
IRo
IR,
600
15 100 200
23.5 30.9 31.5
31.0 32.0 32.1
0.0 13.6 14.9
30.8 32.1 32.3
0.0 2.9 5.5
29.0 32.0 32.2
1,200
15 100 200
27.8 31.5 31.8
31.8 32.1 32.1
7.5 14.9 15.5
31.6 32.2 32.4
0.0 5.5 6.8
31.0 32.2 32.4
12,000
15 100 200
31.7 32.1 32.1
32.1 32.2 32.2
15.3 16.0 16.1
32.2 32.3 32.4
6.0 7.9 8.0
32.3 32.4 32.5
rates
Some examples of the relative values of the critical inflation rates for differing situations are shown in Table 1. The values given are in terms of inflation on an annual percentage basis, 12 Lr/C, denoted by IR in the table. The situation is for a one year planning horizon with on average two price jumps per year. The ordering cost, A, is set at 5 and annual inventory fraction holding charges, I, of 0.15 and 0.30 are considered. Price jumps can only occur to apply at the start of a month, considered equal to four weeks for the analysis. Annual demands of 600, 1,200 and 12,000 units per year and base prices of 15, 100 and 200 are illustrated.
IRo
7.5 14.8 15.5
(14)
inflation
IR,
15 100 200
Month (L=l) (N-L)ICD-A
IRO
Week (L=4)
600
i.e. m,[l-exp(-LrI)l=
Ordering period Month (L=l)
(13)
Thus the expected total cost of an initial order of (N- L)D plus an order D at the start of the last period is the sum of expressions ( 12 ) and ( 13 ) . The critical inflation values ml and y1 are where this is equal to the total cost of an order for (N-L+ 1 )D, covering the whole horizon, given by expression ( 11). Equating these and combining terms together shows that
C
For only one ordering opportunity per month, L = 1, the two critical inflation rates, m. and ml, are virtually identical as the annual demand and base price increase. Although there may be several different optimal policies for values of m between them, the difference in total cost is so negligible that they can be ignored. Hence the results simplify to the alter-
405
native policies of buying the rounded EOQ or a single purchase covering all demand over the planning horizon, depending on whether m (or) ( mo+ ml ) /2. This is because the EOQ is small and the necessary rounding up to a month’s demand excludes many of the possible policies for low values of m, as they are less than this. By contrast, the lower critical level with weekly ordering is zero, showing that any inflation at all has an immediate effect. An important difference with the models reviewed in Section 1, is that the annual inflation rate has to be about 7.5% above the annual inventory holding fraction charge before one single order is optimal. In those models the single purchase was best for annual inflation rates slightly under the inventory charge. The savings achieved by using the optimal policy derived from the model rather than the classical ordering policy, assuming nil inflation, expressed as a percentage of the total variable costs, are given in Table 2. The total variable costs are all costs excluding the cost of buying the demand over the planning horizon at the base starting price C. The magnitude of the savings depends upon the number of orTABLE 2 Policy percentage savings on EQO ordering c
I=O.l5 Inflation rate
z=o.30 k 1
Inflation rate 2
4
k
1
2
4
15 10% 20% 30%
0% 5% 6% 10% 6O/o 12O/o 12% 30% 35% 39% 40% 60%
0% 1% 5% 0% 6% 9% 45% 48% 52%
100 10% 20% 30%
0% 1% 1 10% 3% 7% 10% 30% 34% 37% 30% 60%
0% 0% 2% 0% 6% 6% 44% 47% 49%
200
0% 0% 3% 10% 3% 7% 9% 30% 33% 37% 39% 60°h
0% 0% 1% 0% 3% 5% 44% 47% 49%
10% 20% 30%
dering opportunities per month. They are quite substantial for weekly or fortnightly ordering. 6.3 The importance process
of the inflation rather than the rate alone
The annual inflation rate is a result of price jumps of size m occuring at an average rate of Yper unit of time. The resulting annual inflation rate is equal to 12 Lmr/c. However alternative values of (my) giving the same annual inflation rate do not necessarily‘have the same ordering policies. This would only be true if they gave the same expected price at each ordering opportunity. Let two such cases be (m,r) and (m*,r*). The probabilities of having one price jump in the first month are [ 1 -exp( -kr)] and [ 1 -exp( -Lr*)]. Hence, assuming the same base starting price C, the expected prices at the start of the second month are C+m[ 1 -exp( -Lr)] and C+ m* [ 1- exp ( - Lr* ) 1. These values are equal where M*=WZ{ 1 -exp(
-Lr)}/{
1-exp(
-Lf)}
This equality will also imply that the respective expected market prices at the start of any future time period will be the same. It also ensures that the same values for m. and m, apply to the two situations. An important consequence of the above analysis is that it is the manner in which the inflation occurs that determines the optimal ordering policy, not just the resulting annual inflation rate. Two situations with the same annual inflation rate, i.e. the same value for the product mr, do not in general have the same optimal ordering and stockholding policy. 6.4 A standardised policy calculation
inflation
situation
for
The above result shows that the situation with values (m,r) has the same optimal ordering policy as the situation with values (m' , 1 / L) , where
406
m{1 -exp( mu’=
-Lr)}
{l-exp(-l)}
The case (m’, 1/L) means a price jump of size m’ each month on average. We only need to solve the model for a particular set of operational parameters, demand, inventory cost and set-up cost, to find a standard optimal ordering policy as a function of m =and Y = 1/L. We can then easily find the solution for any general m and r by looking up the standard optimal policy for its equivalent m’ . 6.5 Standardised inflation levels
formulae
for the critical
In the analysis above, it was more convenient algebraicly to define the parameters in terms of the basic time between ordering opportunities. It is necessary to convert these to the standard notation used in inventory control theory. The values IC and D are now the unit inventory holding charge and demand per year. Also let R be the expected number of price jumps per year. The critical value for m and R are now given by DIC-
matl-exp(-R,/12)l=D(0,51+12L)
144 L=A
7. CONCLUSIONS The analysis described has shown that the effect of price inflation in the item purchased has a complex effect on the optimal ordering and stock holding policy. The results given in Table 1 suggest that the upper critical level for annual inflation, IR,, does not vary significantly for large ranges of value in the operational parameters. A reasonable approximation would be to take 16.25% or 32.25% as the inflation rate above which a single purchase covering all requirements is made at the start
of the planning horizon, for annual inventory holding rates of 15% and 30% respectively. Note that his is higher than the inventory holding rate, by about 7.5%, in contrast to Buzacott’s and Naddor’s models where the single purchase policy applied at inflation rates below the inventory holding rate. Thus as with those models, the renewal theory model does not provide a precise answer for very high inflation rates. The problem in these circumstances has been transformed from an inventory planning problem to a strategic buying problem. It is to determine the appropriate length for the planning horizon, which gives the size for the single purchase. This depends upon other factors, including the length of time the high inflation rate is expected to continue, how far ahead a company is willing to commit itself, its attitude to the value of large capital investments in purchased materials and the time it expects to continue making the final product using those materials. It becomes a strategic investment decision. One possible approach is to discount future expenditures and incomes to allow for the changing value of money over time. Note that the discount rate will be the general level of inflation in the economy, not the inflation rate of the material being purchased, which may be due to effects external to the economy. This is being explored currently. The model assumes that the cost of capital in the stockholding cost is not affected by inflation. During the high inflation of the late seventies, interest rates did not increase significantly in line with general inflation, indeed in many countries for several years interest rates were below inflation rates giving a negative real return on capital invested. Furthermore, profits of companies did not rise with inflation either. Thus whatever approach is used, the opportunity cost of capital did not in practice vary significantly as inflation increased. In recent years, however, some governments have used increased capital interest rates to control
407 inflation, so evaluating the model with a direct link between capital charges and inflation is a useful line of research. A final area of possible research is to examine the conflicting interest of buyers and sellers in terms of the best pricing policies, for example large jumps in price at longer time apart compared to small jumps more frequently. REFERENCES I
Buzacott, J.A., 1975. Economic order quantities with inflation. Opl Res. Q., 26(3): 553-558. 2 George, T., 1977. Impact of inflation on financial state-
ments: implications for investors and managers. Unpublished Ph.D. Thesis, Case Western Reserve University, Cleveland, U.S., pp. 89-128. 3 Jagieta, L. and Michenzi, A.R., 1982. Inflation’s impact on the economic order quantity (EOQ) formula. Omega, lO(6): 698-699. 4 Naddor, E., 1966. Inventory Systems. John Wiley. 5 Lev, B. and Soyster, A.L., 1979. An inventory model with finite horizon and price changes. J. Opl Res. Sot., 30: 4353. 6 Gee, K.P., 1977. Management Planning and Control in Inflation. Macmillan Press, London, pp. 70-94. 7 Neuts, M.F., 1973. Probability. Allyn and Bacon Inc., Boston, pp. 180- 183 and 322-329. 8 Boussotiane, A., 1988. Ordering and stockholding under price inflation. Unpublished Ph.D. Thesis, University of Lancaster.
395
I 7 ( 1989) 395-407
Elsevier Science Publishers B.V., Amsterdam -Printed
in The Netherlands
ORDERING AND STOCKHOLDING UNDER PRICE ~N~~AT~ON WHEN PRICES INCREASE IN SUCCESSIVE DISCRETE JUMPS* Brian G. Kingsman and Aziz Boussofiane Manufacturing Systems Research Group, Department of Operational Research and Operations Management, The Management School, University of Lancaster, Bailrigg, Lancaster LA 1 4 YX (Great Britain)
ABSTRACT
.l/last purchased materials and supplies are subject to increases in price over time. Thus a vitul asszlmption qf classical inventory theory, t~~)?~.~ta~t unit prices, is incorrect. Prev~o~~~~ work in this area has ass~~~~~ed either ~ontin~~o~~s esponentialgrowth in prices or that the data qfthe ne_u price rise is known in advance. Investigation showy thut pricesproceea’ via a series of discrete jumps ,~~)~~~)~,ed by plateaus. The time be-
tween successive jumps is stochastic. This paper makes use of‘some analogies with renewal theory to produce appropriate ordering and stockhording poIicies. The model and some interim results are described. There are two critical Ievels for inflation, one below which inflation plays no role in the determination of the order quantities, and a second above which a single purchase stralegj? is best.
1. INTRODUCTION
ilar inflationary situation, to a greater or lesser extent, occurred in all developed economies over this time. Because of this, suppliers became reluctant to guarantee prices for much longer than three months. A basic assumption of classical ordering and stock control theory, constant prices for the indefinite future, was thus broken. Yet, the effect of price inflation on purchasing and stockholding policies has apparently received very little attention in the published literature. Governments changed policies to attempt to bring inflation under control. In the U.K., inflation in industrially purchased materials fell to zero between 1977 and 1978, then rose to 10% per year and remained around this level until 1984, see Fig. 1. Although prices fell since then, there is a continuing possibility of high
Prior to 1970, price inflation in developed economies was small. typically around 4% per annum or lower, so buyers could rely on the prices being offered for component parts, manufactured and semi-manufactured materiais being stable for periods of a year ahead and longer. The increases in price were usually quite small. From 1973 to 1977, inflation for industrially purchased materials in the U.K. was running at an annual rate of 25-300/o, see Fig. 1, which shows the annual values of the U.K. Index for Materials and Fuel Purchased by British Industry from 1966 to 1987. A sim*Presented at the 5th International Working Seminar on Production Economics, I&, Austria, February 22-26, 1988.
0167-188x/89/$03.50
0 1989 Elsevier Science Publishers B.V.
396
300
200
100 tJ 0’ 64
” 66
Iear ‘1
68 70
”
72
74
‘1’
76 78
8002
Fig. 1. Index of prices of U.K. industrially rials 1964-1987 (base of 100 in 1970).
0
’
84
86
purchased
mate-
levels of inflation reoccurring. In developing countries high inflation is apparently endemic and has been so for many years. Inflation rates of over 100% per annum are not uncommon occurrences. So exploring the effect of price inflation on ordering and stock control theory remains an important practical problem. Buzacott [ 1 ] extended the classical EOQ to cover this situation, assuming the same exponential growth in all the inventory cost components as in the cost of the purchased material. To a very close approximation, he showed that the optimal purchasing and ordering policy is the same as the classical EOQ but with a modified inventory fraction holding charge, I*, which depends on the final product pricing policy used by the manufacturing company buying the material. If the final product price is independent of the purchased materials’ cost, then I* is the usual value minus the annual inflation rate. However, if the inflation rate is larger than the cost of holding stock, as it was for the U.K. in 1973-1977 and in the U.S. for some years, the model has no solution. All that can be derived is one initial purchase of an in-
finite or indefmite quantity of material. If the final product price is a fractional mark-up on total costs, ( 1 + k), then I* is the normal value plus k times the inflation rate. This gives more frequent purchases of smaller quantities than the classical EOQ. The motive behind this is to adjust one’s prices rapidly and frequently to reflect inflation. This would imply that the profits of industry should generally have maintained their value over time. The fact that this was not true for British industry, for example, over these years, is probably an indication that this is not a final product pricing probably in general use. Finally. if the final product price is a constant mark up on the materials cost, as occurs in some sectors of the non ferrous metals industry, then 1* is the normal inventory holding fraction charge. The same assumption of how inflation occurs was used by George [2] and Jagieta and Michenzi [ 31. Their results were very similar to those of Buzacott [ 11. The assumption of a continuous exponential growth in the unit price over time appears reasonable on a price index basis at times, see Fig. 2 which shows the same index as Fig. 1 on a monthly basis for the high inflation period, 1972 to mid 1977. As can be seen, the index values fit very closely an exponential growth curve lndex=92.76exp(O.O213T)
where T= 1 in January 1972, 2 in February 1972, etc. The temporary surge above the growth curve in late 1973 was the effect of the fourfold increase in oil prices following the Yom Kippur Arab Israeli war. Whilst possibly true for an index of materials, the assumption of a continuous exponential growth in prices is not valid for individual materials. Suppliers prefer to provide more stability in their prices by making step changes in the price at infrequent intervals of time apart. Some examples of the actual prices charged by suppliers over the high inflation period are given in Fig. 3, for four products
397
Fig. 2. U.K. Index of materials and fuel purchased by manufacturing
7rJrJ _
100
industry.
Unit Price in f per tonne
-
0
1
1973
I
1974
I
1975
1977
Fig. 3. Examples of inflation on prices of individual materials.
bought by an industrial company for use in producing a final product. In late 1973 and early 1974, when high inflation was a new and strange phenomenon, there were frequent small price increases in some cases. Thereafter, suppliers introduced more stability into their pricing. Thus as well as not giving a solution in many circumstances, Buzacott’s model is based on a false picture of how inflation affects the prices of individual materials. Often the step
increases in price were similar over time for a particular product. What seemed to change was the time between successive increases. An alternative approach has been taken by Naddor [ 41 which partially fits the circumstances of step changes in price. He considered what quantity to buy just prior to a known single step price increase, known both in size and timing. Let c and c* be the old and new prices, D the mean annual demand, I the inventory
398 holding fraction charge and Q and Q* the Wilson EOQ values for the old and new prices. The solution is to continue to buy in lots of size Q until the day prior to the price increases. On that day a purchase should be made of an amount
Once this quantity is used up, the material should be purchased in lots of size Q*. The Naddor model was largely an academic exercise. It has been further developed by amongst others Lev and Soyster [ 5 1. Their emphasis was on determining better what orders to place from the date the price rise is announced to the date the rise actually occurred. However, the above model still plays the central role in calculating the size of purchase to make just prior to the price rise. Gee [ 61 also considered this model from an accountant’s viewpoint, on its implications for cost control procedures and the measurement of purchasing and stockholding performance. If there is one price rise per year and the inflation rate equals the inventory holding fraction charge, then (c*-c)/ (Zc) is equal to 1. Thus the recommended order quantity is the annual demand, D, plus Q* (c*/c). This quantity covers the demand over the period to beyond the second price rise. Hence the quantity should be calculated to cover two price rises, which by the same analysis will give a quantity covering demand up to beyond the third price rice, etc. The continuation of this analysis shows that when the annual total price rises are larger than the annual inventory holding rate, or more accurately a value just under the inventory holding rate, the amount to purchase just prior to the first price rise is an indefinite infinite quantity. Thus just like the Buzacott Naddor’s model fails in these model, circumstances. More importantly, Naddor’s model explicitly assumes that the date and size of the price rise are known well in advance. Advance notice of a price rise is sometimes given, possibly
up to a month in advance. An unstated but vital assumption is that the supplier will allow such a very large purchase, covering several months’ usage requirements, at the old price just prior to the day of the increase followed by a long period of no orders. It is rather unlikely that the supplier would accept such an order as it is completely contrary to his intentions in increasing his prices. If there are many suppliers and the consuming company randomly varies its purchases between them as a normal policy, it may be feasible. In the more general case of only one or two suppliers, they will have a pretty good picture of their customer’s usage pattern and so know the buyer is trying to take undue advantage of their advance warning of a price increase. Even if a larger order than normal is accepted by the supplier, he is likely in the future to react to the disadvantage of the buyer by increasing his prices earlier and by larger amounts than he would normally have intended. Clearly an ordering model is needed that anticipates possible price rises in advance. 2. A RENEWALTHEORY PRICE INFLATION
APPROACH
TO
From the examples of Fig. 3, the most realistic model of how price inflation occurs is as a series of step jumps followed by different plateau levels. It seems reasonable from the examples to assume that the size of the step increase is roughly constant, either in absolute or percentage terms. However, the occurrence and timing of the price increases are unknown. It is assumed that the time between successive price increases follows some known probability distribution, calculated from past data. To determine the best future ordering policy, requires estimates of the probability of having a certain price at any time in the future, knowing the past history of price changes. More specifically, since prices increase in a series of discrete jumps, it is necessary to know whether the price has increased since the pre-
399 vious purchasing opportunity and by what amount. The expected price at time t will be the base price at time zero plus the expected number of price jumps that have occurred in the interval [ O,t]. This requires the calculation of the distribution of the number of arrivals (price jumps) in the time interval [ O,t], where the number of arrivals will be integer, and the probability of having a price P, say, at time t+ At knowing that there have already been n price jumps by time t. The emphasis is on the number of price jumps and in the time between jumps. This requires a renewal theory approach rather than a queuing theory approach. We can draw the analogy between price jumps and machine breakdowns and between the time between two successive price jumps and the service time or life time of a machine. The probability of having a price P at time t+ At knowing the price was P at time t is equivalent to calculating the probability of having a machine in service at time t+ At knowing it was already in service at time t. This analogy means that we can directly use some of the standard results in renewal theory, e.g. Neuts [7]. Two probability distributions have been used to model the time between price jumps. The first simple case is a negative exponential distribution. Observations on a very limited data sample suggested that this did lit some cases. Furthermore, the calculation of the prices expected at particular future times is enormously simplified because of the ‘memoryless’ property of this distribution. This means that the probability of any price change does not depend upon what happened before, so all that needs to be considered is the probability of no arrival during an interval of time t. This is the one used to derive the results given in the final section of this paper. The second distribution is the Erlang distribution. This is a very general distribution enabling several different models of the inflation process to be evaluated. In particular, the probability of the next price jump occurring
can be made dependent on the time since the last jump occurred. It seems intuitively likely in practical situations that the probability of a price jump taking place will increase as the time elapsed from the last jump increases. Since the density function is a Gamma distribution, the calculations of the expected prices are again considerably simplified because of the standard well established properties of Gamma distributions. This analysis is continuing and will be described in later papers. 3. THE PURCHASING AND STOCKHOLDING MODEL The ordering and stockholding model is formulated for the case of a general inter-arrival time distribution. The objective is to minimise the costs of purchasing the known requirements over some planning horizon. The planning horizon is divided into N equal time intervals, e.g. months or weeks, such that there cannot be more than one price increase per interval. The demand for the item purchased is assumed to occur at a constant rate D per period. All demand must be met on time, so no shortages or backorders are allowed. A decision on whether to make a purchase or not is made at the beginning of each time interval. The problem is clearly a sequential decision making situation which can naturally be formulated in dynamic programming form. The decision at future time periods will depend upon the price prevailing then plus the amount of material in stock. Let fx(S,p) be the expected minimum costs of meeting the requirements over the remaining k periods of the planning horizon with a current stock of S units of material and a current price of p per unit. Let Po( k,n), where n < N- k, be the probability of having no increase in price in period k when there have already been y1price increases before period k. The price will then remain at p for the start of the next period k- 1. The probability of a price increase occurring is 1 -PO (k,n), whence the price at the start of the next period k- 1 will be
400 p+)??, where m is the jump in price. If a purchase is made to raise the stock on hand to Y,, then the resulting costs will be r;(S,p,~~‘,)=Sc(S,Y,)fpuR(S,~.y,) +INV(.~.a.Y~)+p,,(k,n).~~,(Y~-D,p)
+(l-P,,(k,n)If;~,(Y~-D,P+m)
(1)
where SC, PUR and INV are the set-up, purchasing and inventory holding costs incurred in period k. Then .~(S,P)=
Min
1, .I>.,; .\
.f,(S,p,Y,)
(2)
The set-up cost is an internal cost to the inventory/purchasing system. It will not normally be subject to the same inflation as in the price of the materials bought, particularly if these are imported. For simplicity, it is assumed to remain constant at a value A. Hence SC(S,Y,)=,-l =0
if Y,>.S otherwise
There is, however, no conceptual difficulty for the model in allowing the set-up cost to vary over time if there is internal inflation in the system. The purchasing cost is merely the amount purchased times the current price at the start of period k. Since the unit cost is p PUR(S,~.Y,~)=~(J,~-S) =o
if Yk;>.S otherwise, i.e. no purchase
made
The standard inventory holding cost model is that the cost incurred by a stock quantity, x say. is the average level of stock over the period until the s is used up in meeting demand times the cost of holding one unit of stock per unit time. This latter factor is taken as some inventory holding fraction charge times the unit cost of the item. Usually in dynamic programming inventory models, it is assumed that inventory holding costs payments are made regularly each period on the average amount of
stock hold over the period or the end of period stock. This is costed on the prevailing unit price of the item at the start of the period. Prices increase over time under the stochastic process specified. Although the value of the stocks held increases after an increase in price, the actual amount of money required to finance the stock remains the same. The same effort is required to look after the stock. The only factor that changes is the insurance component in the inventory holding charge. This is small and in most companies is only reviewed and changed once or twice per year. Hence the inventory holding cost should be calculated on the basis of the price paid for each unit of stock. This would require identifying each component of the opening stock of a period according to the price paid, thus requiring many state variables for the units of stock bought at the differing prices. This would make the DP (dynamic programming) model very complex and difficult to update. An alternative approach is to assume that the whole of the inventory holding cost arising from a purchase in period k is incurred in period k. This does not mean that the actual inventory charges are paid out instantly in period k. We are simply including the total inventory expenses that will be incurred in period k and future periods in period k rather than spreading them out over periods k to the time that order is used up, as is the usual approach. If, at time t, the stock in hand is s and a purchase is made at price p to raise the stock to y, this purchase will be used to meet demand over the period t + s/D to t + y/D, where D is the demand per period, see Fig. 4. The inventory holding costs incurred by the stock s have already been included in the model at earlier periods. Hence the total inventory holding cost incurred by the purchase y--s will be given by area (2) of Fig. 4, (y’-s2)/2D, times Ip. Hence, the inventory holding costs for period k to use in the model are INV(S,p,Y,)
=Ip/2D(
Y, ‘-4’)
(3)
401 The mean value 1/Y will represent the average time between price jumps, hence Yis the average arrival rate, number of price jumps per unit time. If X is the random variable representing the (continuous) time until the next price jump, then
Stock
Prob{X>t}=
1 -F(t)=exp(
-rt)
The ‘memoryless’ property is the so-called Markov property that for every t> 0 and x> 0 Prob{Xzt+x(X>t}=Prob{X>x}=exp(-rx) Time Fig. 4. Inventory costs.
An implicit assumption in this model is that stock is used up on a FIFO basis. The later stock may be bought at a higher price thus incurring higher inventory charges. It would thus be cheaper to use up the higher priced stock first. This implies that the inventory system is being managed to make speculative gains from the stocks held. This goes against the common practice of most companies. The gains from such a policy are likely to be small. It would make the DP model extremely complex as it would have to include all the possible delivery alternatives. An increased number of state variables would be required to keep track of the stocks bought at different prices. Since the gains are small, as mentioned, and the modelling difficulties large, this aspect is ignored and a FIFO use of stock therefore assumed. 4. NEGATIVE EXPONENTIAL TIME DISTRIBUTIONS
ARRIVAL
If a variable x follows a negative exponential distribution with mean 1/Y and variance 1 /r2, then its density function is given by f(x)=rexp(-rx)
forx>O
=o
for x-z0
The cumulative
distribution
F(x)=Prob{X
forx20
=o
forx
function
of X is
(4)
This is easily shown to be true for this negative exponential distribution. The ordering and stockholding model requires the value of PO(k,n ), the probability that no price jump takes place in period k, given that n price jumps have occurred previously. The ‘memoryless’ property means that this probability is independent of the previous number of price jumps, ~1,and the particular point in time reached, i.e. the period k. It is equal to the probability that there is no jump in price over the length of time in the period k. Since time has been divided into equal time intervals, the length of these epochs can be taken as the basic unit time. Hence putting x= 1 in (4) Po(k,n)=Po =Prob{X>
1)
(5)
= exp( -r)
5. SOLUTION
TO THE DP MODEL
The above value for PO can be substituted into the DP model. Unfortunately, it is not possible to determine a simple analytical solution to the model. It is necessary to solve the model numerically for particular sets of values for the various parameters. The solution to the DP model gives the optimal solutions at the start of each period as a function of the stock on hand at that time and the prevailing price. As might be expected the optimal policies are rather complex. A number of cases have been analysed and will be discussed in Boussofiane
[81. The major
problem
with all dynamic
pro-
402
gramming models is the large amount of computational effort required to find these solutions, even for quite small planning horizons. This can be reduced by increasing the length of the time epoch between successive price jumps. This makes sense for practical application since the time between jumps is likely to be months rather than days. Thus allowing one price jump per month is reasonable. However, the general model of Section 3, eqns. ( 1) and (2 ), and the derivation of PO, (5 ), assumed one ordering opportunity as well as one price jump per time epoch. Ordering opportunities more frequently than once per month are more appropriate in practical situations. This can be achieved by allowing L ordering opportunities between the times at which price jumps can occur. The basic unit time period is the time between ordering opportunities. Hence the time x in eqn. (4) now has a value of L time units, so the probability a price jump occurs is now P,,(k,n)=Pq
=Prob{X>Lj
(‘5)
=exp( -Lr) where l/r is the expected number of periods between successive price increases. If the arrival of price jumps is constrained to be only once per month at most, and there are L ordering opportunities per month, then the expected number of price jumps per year will be 12 Lr. The expected price increase over the year will thus be 12 Lrm. 6. SOME
GENERAL
RESULTS
However, it is possible to derive some general properties of the solution policy for the case of a negative exponential distribution for the time between price jumps. 6.1 The critical
inflation
rates
In the situation analysed, time is divided into discrete periods with an ordering opportunity at the beginning of each period. Since no short-
ages are allowed, the order quantity must cover at least the time until the next ordering opportunity. This time interval is taken as the basic unit time period. For simplicity at this stage of the analysis, all parameters are defined in terms of this unit time, so IC and D are the unit stockholding charge and demand over one period between successive ordering opportunities. It is assumed that if there is no inflation, i.e. the price jump m is zero, then the optimal order quantity, the rounded EOQ will cover the demand over one period. This will be the case if A < ICD
(7)
In this situation, the optimal order quantity will be to order 1 period’s demand at the start of each future period. Larger orders will only lead to increased stockholding costs. As inflation increases above zero, then there will come a point where it will be worth carrying stock to avoid a price increase in subsequent periods. In the general situation, no price increase can occur before the start of period L. Clearly, there is no advantage in buying other than a period’s demand at each of the first L - 1 ordering opportunities. The effect of inflation will be to increase the orders from the start of period L- 1 onwards. The initial impact of increasing inflation, therefore, affects the ordering decisions over the periods L- 1 and L, whether to have two separate equal orders or one order for the two periods. Let the arrival rate and size of the price jump where this occurs be denoted by r. and m,. The stock controller is indifferent between one order covering 2 periods’ demand and two orders each of 1 period’s demand, where the second of these orders might have to be bought at a higher price. Letting the base starting price be C per unit, the total costs incurred by an order of 20, covering 2 periods’ demand, placed at the start of period L- 1 will be 2 CD+A+O.S
IC(4 D)
(8)
403
The cost of an order of size D, covering only one period’s demand at time L- 1 will similarly be CD+A+Q.SlCD
If no price increase has occurred during the first month, then the second order of size D placed at the start of period L will have exactly the same cost. However, if a price increase has occurred, the unit cost of the items in the second order will be C+mo, hence the total costs of the second order will be (C+mo)D+A+0.5Z(C+mo)D
(9)
The probability of no price increase is exp ( - Lr, ), see eqn. (6 ), the probability of an increase 1 - exp ( - Lr,) . Multiplying expressions ( 8 ) and (9 ) by the probabilities of each event occurring, the expected cost of the two order policy simplifies to 2 CD+2 A+ZCD+{
1 -exp(
-Lr,,)}{rq,D+O.S
ImoD}
This will be the same as the cost of the one order covering 2 periods, expression (7) if DIC-A
mO[1-exp(-Lr0)l=D(0.51+1)
(10)
Note, that because of the condition given by eqn. (7) m, is positive or zero. As inflation increases then clearly it is worthwhile buying even more initially in period L- 1 at the base price. The precise effect on the order sizes depends upon the length of the planning horizon, the number of periods over which it is desired to minimise costs. However, there are many alternative possible sequences of orders so it is impossible to work out a simple solution. Analysis of a number of examples suggests that as inflation increases the number of orders placed to cover any given horizon decreases, but not in a simple way. For example, with a 12 period horizon there could be several alternative optimal ways of having only 4 orders, depending on the value of the price jump m. Finally, there must come a point when infla-
tion is so high that it is optimal to buy the whole remaining requirements for the planning horizon in period L- 1. Consider an N period planning horizon. The stage just prior to placing an order for (N-L + 1 )D, covering the remaining horizon, is to place an order covering the first N-L periods followed by1 order of size D covering the last period’s demand. The total costs for one order of size (N-L+l)Dwillbe C(N-L+
l)D+A+0.5
ZC(N-L+
1)2D
(11)
and similarly for an ordering covering N-L periods’ demands, the total costs are C(N-L)D+A+O.S
ZC(N-L)2D
(12)
The cost of the final order, D, at the start of the last period will depend upon the price at the start of the last period. Let P, be the probability the price is C+jm at the start of the last period. If the planning horizon is a multiple of L, the minimum interval between price jumps, then the maximum possible number of price increases over the planning horizon is (N / L- 1). If not, then the maximum number of possible price increases is the integer less than (N/L- 1). Hence for generality let K be the maximum number of price increases possible over the planning horizon. Thus the expected cost of the order size D at the start of the last period is ,to { (C+jm)D+A+0.5 CD+A+O.S
ICD+mD(
Z(C+jm)D}P,= 1+0.51)
F jP, ,=L
Since the maximum number of price jumps possible is K, the probability the price is C+jm is the probability there have been exactly j occasions in which an arrival has occurred and exactly K-j with no arrivals. The probability of j arrivals is [ 1 - exp ( - Lr) ]j and K-j non arrivals is [ exp ( - Lr) ]“-I. We are not inter-ested in the precise arrangement of arrivals and non arrivals merely the total number of arrivals. There are “C, permutations ofj arrivals
404
in the maximum
TABLE 1
possible K jumps.Therefore
Examples of the critical inflation rates for differing ordering opportunities per price epoch
Prob (price = C+ jm) = P, =KC,exp(
[ 1-exp(
-Lr(K-j)
-LT)]’ (a) Annual inventory holding fraction charge of 0.15
This is a binomial distribution with parameter exp( -Lr). The final term in eqn. ( 11) is just the mean value of this binomial distribution, which is K[ 1 - exp ( - Lr) 1. Thus the cost of the final order simplifies to CD+A+O.S
CD+mD(
1+0.5 I)K[ 1 -exp(
-Lr)]
Annual demand
0.5 IC(N-L+l)2D=A+0.5
IC(N-L)‘D+0.5
+m,D(1+0.5
ICD
I)K[l-exp(-LLr,)]
6.2 Examples
Fortnight (L=2)
IRo
IRo
KD(1+o,51)
of the critical
IR,
15.0 16.0 16.1
0.0 5.5 6.8
14.7 16.0 16.1
0.0 0.0 1.5
13.0 15.8 16.0
1,200
15 100 200
12.0 15.5 15.9
15.8 16.1 16.2
0.0 6.8 7.5
15.4 16.1 16.2
0.0 1.5 3.4
14.0 16.0 16.1
12,000
15 100 200
15.8 16.1 16.1
16.2 16.2 16.2
7.2 8.0 8.0
16.2 16.2 16.2
2.3 3.8 3.9
16.0 16.2 16.2
(b) Annual inventory holding fraction charge of 0.30 Annual demand
C
Ordering period Fortnight (L=2)
Week (L=4)
IR,
IR,
IRo
IR,
IRo
IR,
600
15 100 200
23.5 30.9 31.5
31.0 32.0 32.1
0.0 13.6 14.9
30.8 32.1 32.3
0.0 2.9 5.5
29.0 32.0 32.2
1,200
15 100 200
27.8 31.5 31.8
31.8 32.1 32.1
7.5 14.9 15.5
31.6 32.2 32.4
0.0 5.5 6.8
31.0 32.2 32.4
12,000
15 100 200
31.7 32.1 32.1
32.1 32.2 32.2
15.3 16.0 16.1
32.2 32.3 32.4
6.0 7.9 8.0
32.3 32.4 32.5
rates
Some examples of the relative values of the critical inflation rates for differing situations are shown in Table 1. The values given are in terms of inflation on an annual percentage basis, 12 Lr/C, denoted by IR in the table. The situation is for a one year planning horizon with on average two price jumps per year. The ordering cost, A, is set at 5 and annual inventory fraction holding charges, I, of 0.15 and 0.30 are considered. Price jumps can only occur to apply at the start of a month, considered equal to four weeks for the analysis. Annual demands of 600, 1,200 and 12,000 units per year and base prices of 15, 100 and 200 are illustrated.
IRo
7.5 14.8 15.5
(14)
inflation
IR,
15 100 200
Month (L=l) (N-L)ICD-A
IRO
Week (L=4)
600
i.e. m,[l-exp(-LrI)l=
Ordering period Month (L=l)
(13)
Thus the expected total cost of an initial order of (N- L)D plus an order D at the start of the last period is the sum of expressions ( 12 ) and ( 13 ) . The critical inflation values ml and y1 are where this is equal to the total cost of an order for (N-L+ 1 )D, covering the whole horizon, given by expression ( 11). Equating these and combining terms together shows that
C
For only one ordering opportunity per month, L = 1, the two critical inflation rates, m. and ml, are virtually identical as the annual demand and base price increase. Although there may be several different optimal policies for values of m between them, the difference in total cost is so negligible that they can be ignored. Hence the results simplify to the alter-
405
native policies of buying the rounded EOQ or a single purchase covering all demand over the planning horizon, depending on whether m (or) ( mo+ ml ) /2. This is because the EOQ is small and the necessary rounding up to a month’s demand excludes many of the possible policies for low values of m, as they are less than this. By contrast, the lower critical level with weekly ordering is zero, showing that any inflation at all has an immediate effect. An important difference with the models reviewed in Section 1, is that the annual inflation rate has to be about 7.5% above the annual inventory holding fraction charge before one single order is optimal. In those models the single purchase was best for annual inflation rates slightly under the inventory charge. The savings achieved by using the optimal policy derived from the model rather than the classical ordering policy, assuming nil inflation, expressed as a percentage of the total variable costs, are given in Table 2. The total variable costs are all costs excluding the cost of buying the demand over the planning horizon at the base starting price C. The magnitude of the savings depends upon the number of orTABLE 2 Policy percentage savings on EQO ordering c
I=O.l5 Inflation rate
z=o.30 k 1
Inflation rate 2
4
k
1
2
4
15 10% 20% 30%
0% 5% 6% 10% 6O/o 12O/o 12% 30% 35% 39% 40% 60%
0% 1% 5% 0% 6% 9% 45% 48% 52%
100 10% 20% 30%
0% 1% 1 10% 3% 7% 10% 30% 34% 37% 30% 60%
0% 0% 2% 0% 6% 6% 44% 47% 49%
200
0% 0% 3% 10% 3% 7% 9% 30% 33% 37% 39% 60°h
0% 0% 1% 0% 3% 5% 44% 47% 49%
10% 20% 30%
dering opportunities per month. They are quite substantial for weekly or fortnightly ordering. 6.3 The importance process
of the inflation rather than the rate alone
The annual inflation rate is a result of price jumps of size m occuring at an average rate of Yper unit of time. The resulting annual inflation rate is equal to 12 Lmr/c. However alternative values of (my) giving the same annual inflation rate do not necessarily‘have the same ordering policies. This would only be true if they gave the same expected price at each ordering opportunity. Let two such cases be (m,r) and (m*,r*). The probabilities of having one price jump in the first month are [ 1 -exp( -kr)] and [ 1 -exp( -Lr*)]. Hence, assuming the same base starting price C, the expected prices at the start of the second month are C+m[ 1 -exp( -Lr)] and C+ m* [ 1- exp ( - Lr* ) 1. These values are equal where M*=WZ{ 1 -exp(
-Lr)}/{
1-exp(
-Lf)}
This equality will also imply that the respective expected market prices at the start of any future time period will be the same. It also ensures that the same values for m. and m, apply to the two situations. An important consequence of the above analysis is that it is the manner in which the inflation occurs that determines the optimal ordering policy, not just the resulting annual inflation rate. Two situations with the same annual inflation rate, i.e. the same value for the product mr, do not in general have the same optimal ordering and stockholding policy. 6.4 A standardised policy calculation
inflation
situation
for
The above result shows that the situation with values (m,r) has the same optimal ordering policy as the situation with values (m' , 1 / L) , where
406
m{1 -exp( mu’=
-Lr)}
{l-exp(-l)}
The case (m’, 1/L) means a price jump of size m’ each month on average. We only need to solve the model for a particular set of operational parameters, demand, inventory cost and set-up cost, to find a standard optimal ordering policy as a function of m =and Y = 1/L. We can then easily find the solution for any general m and r by looking up the standard optimal policy for its equivalent m’ . 6.5 Standardised inflation levels
formulae
for the critical
In the analysis above, it was more convenient algebraicly to define the parameters in terms of the basic time between ordering opportunities. It is necessary to convert these to the standard notation used in inventory control theory. The values IC and D are now the unit inventory holding charge and demand per year. Also let R be the expected number of price jumps per year. The critical value for m and R are now given by DIC-
matl-exp(-R,/12)l=D(0,51+12L)
144 L=A
7. CONCLUSIONS The analysis described has shown that the effect of price inflation in the item purchased has a complex effect on the optimal ordering and stock holding policy. The results given in Table 1 suggest that the upper critical level for annual inflation, IR,, does not vary significantly for large ranges of value in the operational parameters. A reasonable approximation would be to take 16.25% or 32.25% as the inflation rate above which a single purchase covering all requirements is made at the start
of the planning horizon, for annual inventory holding rates of 15% and 30% respectively. Note that his is higher than the inventory holding rate, by about 7.5%, in contrast to Buzacott’s and Naddor’s models where the single purchase policy applied at inflation rates below the inventory holding rate. Thus as with those models, the renewal theory model does not provide a precise answer for very high inflation rates. The problem in these circumstances has been transformed from an inventory planning problem to a strategic buying problem. It is to determine the appropriate length for the planning horizon, which gives the size for the single purchase. This depends upon other factors, including the length of time the high inflation rate is expected to continue, how far ahead a company is willing to commit itself, its attitude to the value of large capital investments in purchased materials and the time it expects to continue making the final product using those materials. It becomes a strategic investment decision. One possible approach is to discount future expenditures and incomes to allow for the changing value of money over time. Note that the discount rate will be the general level of inflation in the economy, not the inflation rate of the material being purchased, which may be due to effects external to the economy. This is being explored currently. The model assumes that the cost of capital in the stockholding cost is not affected by inflation. During the high inflation of the late seventies, interest rates did not increase significantly in line with general inflation, indeed in many countries for several years interest rates were below inflation rates giving a negative real return on capital invested. Furthermore, profits of companies did not rise with inflation either. Thus whatever approach is used, the opportunity cost of capital did not in practice vary significantly as inflation increased. In recent years, however, some governments have used increased capital interest rates to control
407 inflation, so evaluating the model with a direct link between capital charges and inflation is a useful line of research. A final area of possible research is to examine the conflicting interest of buyers and sellers in terms of the best pricing policies, for example large jumps in price at longer time apart compared to small jumps more frequently. REFERENCES I
Buzacott, J.A., 1975. Economic order quantities with inflation. Opl Res. Q., 26(3): 553-558. 2 George, T., 1977. Impact of inflation on financial state-
ments: implications for investors and managers. Unpublished Ph.D. Thesis, Case Western Reserve University, Cleveland, U.S., pp. 89-128. 3 Jagieta, L. and Michenzi, A.R., 1982. Inflation’s impact on the economic order quantity (EOQ) formula. Omega, lO(6): 698-699. 4 Naddor, E., 1966. Inventory Systems. John Wiley. 5 Lev, B. and Soyster, A.L., 1979. An inventory model with finite horizon and price changes. J. Opl Res. Sot., 30: 4353. 6 Gee, K.P., 1977. Management Planning and Control in Inflation. Macmillan Press, London, pp. 70-94. 7 Neuts, M.F., 1973. Probability. Allyn and Bacon Inc., Boston, pp. 180- 183 and 322-329. 8 Boussotiane, A., 1988. Ordering and stockholding under price inflation. Unpublished Ph.D. Thesis, University of Lancaster.