ORDERING DECISIONS UNDER IMMINENT
PRICE INCREASES1
Ömer S. Benli
California State University, Long Beach
[email protected]
ABSTRACT
It is not unrealistic to assume that the opportunity cost of capital, and hence the
inventory carrying charge, is a continuous function of time. On the other hand, price
increases, due to inflation, for example, take place at discrete time points. Ordering
decisions related with this phenomenon are analyzed in the context of a deterministic,
static demand, single-item, continuous review inventory model.
INTRODUCTION
According to Bartmann and Beckmann (1992), J. M. Keynes differentiated three
motives for holding money which can equally apply to carrying inventories:

The Transaction Motive. Since outflows are not synchronized with
inflows, stocks are needed to bridge these discrepancies.

The Precautionary Motive. One must maintain reserve stocks in order to
satisfy the demand while awaiting delivery.
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Journal of Business and Behavioral Sciences, Vol.12, No. 2, Spring 2005.
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
The Speculative Motive. If prices are expected to rise, it pays to keep
stocks on hand.
Most of the literature on inventory theory deals with the first two motives; this
paper is concerned with the third. Examining several of the excellent texts that were
published recently, such as Camm and Evans (2000), Hillier and Hillier (2003), Redner,
Stair and Balakrishnan (2003), and Powell and Baker (2004), one observes that inventory
related issues are only treated with respect to the above mentioned first two motives.
Inflation is one of the main causes of price increases. As stated in Jolayemi and
Oluleye (1993), ``[in] most countries, particularly [in] the developing countries, inflation
rises daily.'' In many countries this results in having the prices of certain goods and
services to be quoted in some stable currency and then converted into local currency
(Arcelus and Srinivasan, 1993). The consequence of this situation for the buyer dealing in
weaker currency, argues Arcelus and Srinivasan (1993), is future price increases
necessitating the placement of special orders while the lower prices are still in effect.
They further state that ``[it] is customary to give prior notice of the magnitude of these
changes and of their effective date.''
The price increases also take place when demand exceeds the supply. In either
case, especially in monopolistic and oligopolistic markets, the timing and the size of
these increases are more or less predictable.
Starting with Naddor (1966), there exists considerable work on the single
announced price change problem (see, for example, Taylor and Bradley, 1985; Lev and
Weiss, 1990; Aull-Hyde, 1992; Arcelus and Srinivasan, 1993.) Effect of the inflation on
prices, on the other hand, is usually assumed to be continuous (Buzacott, 1975), (Datta
and Pal, 1991).
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MODEL AND ANALYSIS
Suppose a buyer operating under a deterministic, static demand, single-item,
continuous review inventory policy is faced with imminent price increases during a finite
horizon [0, T ] . Without loss of generality, assume that the initial (at time 0) and the
ending (at time T ) inventory is zero. The initial unit purchase price, c1 , will be applicable
until time  1 , after which the unit price will be c2 until  2 , and so on. That is, the unit
purchase price is given by,
c(t )  c , 0  t   ,
1
1
c c
, 
 t   , j  2,
j
j 1
j 1
j
, n.
Let the other parameters be,
k
(10
Fixed cost of placing an order ($ per order).
Percentage (opportunity) cost of capital per unit time.
0)β
d
Constant demand rate (units per unit time).
Also assume that inventory holding solely depends on the money tied up in
inventories and no additional cost factors (such as storage, etc.) are applicable. Thus,
keeping one unit in inventory for one unit of time costs c j  dollars when the unit is
purchased during the time interval ( j 1 , j ] . Graphically, the cumulative demand can be
represented as in the following figure.
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Consider a finite horizon, deterministic, static demand, single-item, continuous
review inventory model. It is easy to show that the optimal order sizes will be of equal
size. A straight forward way of determining the optimal number of orders, say m , during
the planning horizon is, first, assuming that m is ``continuous'' and then differentiating
the total cost function with respect to m . Most likely the resulting optimal value for m
will be fractional.
Optimal integer value for m is then found by checking the
neighboring integers of the fractional m in the cost function. This follows from the fact
that the cost function is strictly convex.
In the following analysis, total cost per unit time in a finite horizon model will be
approximated by its infinite horizon version. In other words, closed form of the total cost
per unit time will be obtained using the “fractional” m . The quality of this approximation
depends on the relative magnitudes of m and k .
When an ending inventory is required in a finite horizon model, it can easily be
shown that if the size of the ending inventory is larger than the equal size orders during
the planning horizon, then it is optimal to place equal size orders to cover the demand
during the finite horizon and place an extra order for the ending inventory at the end of
the planning horizon.
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In order to illustrate the approach used in this analysis, consider the simplest case:
initial and ending inventories are zero and there is a single price increase at  1 . So we
have two periods of constant price. Let  j denote the unit cost of the purchased item
including the inventory related unit costs,
 j  c j  (1/ d ) 2kdc j  ,
j  1, 2.
Also let D j denote the total demand during ( j 1 , j ). Then the total cost for the
two-period problem as the function of the ending inventory in period one (i.e. the size of
the last order at  1 ) is
TC ( I1 )   1 D1  k  c1 I1  c1 I12 /(2d ).
Differentiating with respect to I1 and setting the resulting expression equal to
zero gives the minimizing value for the size of the last order size at  1 as
I1* 
 2  c1 2kd
.
 1  c1 c1
It is straight forward to extend this to multiple price periods.
CONCLUSIONS
It is important to emphasize the basic assumption in the above analysis: the
infinite horizon approximation of the finite horizon costs. If the number of orders during
a constant price period is large enough and/or order costs are small, then the
approximation is acceptable. This seems to be the case in large number situations.
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The above analysis suggests the following practical procedure. At any given
period, j , of constant purchase price, place equal size orders to cover the demand until
 j , then at time  j place an order of size,
I1* 
 j 1  c j
 j  cj
2kd
.
cj
When the inventory is depleted, continue placing equal size orders until the next
price increase at  j 1 .
In the case of nonstationary demand, although in principle dynamic programming
recursion can be used, problems arise from the lack of satisfactory closed form
expression for the unit cost per unit time.
An interesting further question is the problem of determining the timing and the
size of the price increases from the seller's point of view. A possible approach might be
similar to the joint optimization of buyer's and seller's objective functions in the quantity
discount models (see for example, Datta and Pal, 1991.)
REFERENCES
Arcelus, F. J. and Srinivasan, G. (1993). “Generalizing the Announced Price Increase
Problem,” Decision Sciences, 24, 847 – 866.
Aull-Hyde, R. L. (1992). “Evaluation of Supplier-Restricted Purchasing Options under
Temporary Price Discounts,” IIE Transactions, 24, 184 – 866.
Bartmann, D. and Beckmann, M. J. (1992). Inventory Control: Models and Methods,
Springer-Verlag.
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Buzacott, J. A. (1975). “ Economic Order Quantity with Inflation,” Operational Research
Quarterly, 26, 553 – 558.
Camm, J. D. and Evans, J. R. (2000). Management Science and Decision Technology,
Southwestern.
Datta, T. K. and Pal, A. K. (1991). “Effects of Inflation and Time Value of Money on an
Inventory Model with a Linear Time Dependent Rate and Shortages,” European
J. Operational Research, 52, 326 – 333.
Hillier, F. S. and Hillier, M. S. (2003). Introduction to Management Science, McGrawHill.
Jolayemi, J. K. and Oluleye, A. E. (1993). “Scheduling Project under the Condition of
Inflation,” OMEGA Int. J. of Mgmt. Sci., 21, 481 – 487.
Lev, B. and Weiss, H. J. (1990). “Inventory Models with Cost Changes,” Operations
Research, 38, 53 – 63.
Naddor, E. (1966). Inventory Systems, Wiley.
Powell, S. G. and Baker, K. R. (2004). The Art of Modeling with Spreadsheets, Wiley.
Render, B., Stair, R. M., and Balakrishnan, N. (2003). Managerial Decision Modeling
with Spreadsheets, Springer-Verlag.
Taylor, S. G. and Bradley, C. E. (1985). “Optimal Ordering Strategies for Announced
Price Increases,” Operations Research, 33, 312 – 325.
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