Discount strategies for one-time-only sales.

1. Introduction

This paper deals with the problem of temporary over-stocking, a common if undesirable occurrence in most business concerns. When it happens, firms often resort to a wide variety of trade deals to correct the deficiency. The most common takes the form of a discount in

the purchase price of the merchandise to encourage larger than ordinary orders. Such incidences of inventory clearance sales are especially prevalent in cases of older products in highly competitive markets, be they retail or industrial. Further elaboration upon the use of these discounts appears in Blattberg et al. (1981), Blattberg and Levin (1987), Davis and Gaither (1985), Lal (1990), Shipley and Bourdon (1990), Tellis (1986), Wilcox et al. (1987), and in some of the references listed therein. A recent case incidence on t he role of forward buying practices and their effect on supermarket pricing policies appears in The Economist (1992).

The basic temporary-price-reduction problem (Baker, 1976), denoted as TPRP, is well known and appears in many textbooks on the subject (e.g. Tersine, 1988). The simplest version of the TPRP contemplates (i) a vendor announcing a certain grace period within which prospective buyers may place an extraordinary order in exchange for discounts in the purchase price; and (ii) a cost-minimizing buyer attempting to determine the appropriateness of such an offer. Lately, there has been a revival of interest in the TPRP (see, for example, Arcelus and Srinivasan, 1992; Ardalan, 1988; Aull-Hyde, 1992; Tersine and Gengler, 1982; Tersine and Price, 1981), in an effort to adapt it as a solution framework for new applications. It is the purpose of this paper to present a generalized version of the TPRP that explicitly recognizes widely used variations (i) in the length of the grace period; (ii) in the size of the order needed to qualify for the discounts; and (iii) in the portion of the order eligible for the discount.

To that effect, the paper is organized as follows. The basic methodology is presented in Section 2, which includes a full description of the variations listed above, of their implications for the modelling of the TPRP and a discussion of how the existing formulations fit within this general structure. In Section 3 each cost-minimizing objective function is analyzed in detail. Section 4 concludes the paper. All symbols used are listed and defined in Table 1.

Table 1. List and explanation of symbols used

s                  Index denoting the scenarios, s = 1,2,3
c                  Index denoting the cases, c = 1,2,3
P                  Normal purchase price
d                  Discount offered
P'                 Discounted purchase price (P' = P - d)
F                  Holding cost rate per dollar
K                  Ordering cost per order
[t.sub.B]          Beginning of the discount period
[t.sub.F]          End of the discount period
[t.sub.i]          ith, i = 1,..., r, regular ordering point, if
                   any, within the discount period
[t.sub.r]          First (Case 4) regular ordering time during
                   the discount period
[t.sub.f]          Length of the period between [t.sub.r] and
                   [t.sub.F] ([t.sub.f] = [t.sub.F] - [t.sub.r]),
                   for Case 4
[g.sub.as]         Number of orders to be placed, for Case = 4,
                   s = 1,2, alternative a = A,B, during the
                   period between [t.sub.r] and [t.sub.0]
q                  Demand between [t.sub.F] and [t.sub.1], for
                   Case 2, and between [t.sub.1] and [t.sub.F],
                   for Case 3
[Q.sub.as]         Size of the orders to be placed, for Case = 4,
                   s = 1,2, alternative a = A,B, during the
                   period between [t.sub.r] and [t.sub.0]
R                  Yearly demand rate
[Q.sub.cs]         Optimal special order quantity for Case c,
                   Scenario s
[Q.sub.O]          Economic order quantity, when the purchase
                   price is P
[Q.sub.E]          Demand already met with the purchases made
                   at or immediately before [t.sub.F]
[Q.sub.f]          Demand for the period between [t.sub.r] and
                   [t.sub.F] ([Q.sub.f] = [R.sub.tf])
[m.sub.s]          Largest integer not exceeding [Q.sub.f]
                   /[Q.sub.j], [Q.sub.i] = {[Q.sub.0],
                   [Q.sub.1s], [Q.sub.2s]}, for s = 1,2
[[Delta].sub.s]    Change in [Q.sub.s] needed in Case 4 to render
                   [Q.sub.f]/([Q.sub.s] - [[Delta].sub.s])
                   or [Q.sub.f]/[Q.sub.s] + [[Delta].sub.s])
                   greater than or less than [m.sub.s],
                   respectively, for s = 1,2
[n.sub.s]          Largest integer not exceeding [Q.sub.f]
                   /([Q.sub.1s] + [[Delta].sub.s]) for s = 1,2
E([Q.sub.E])       Undiscounted cost of satisfying a demand of
                   R - [Q.sub.E] units
[H.sub.s](Q)       Cost of holding Q units during the [Q.sub.cs]
                   /R period, for s = 1,2
[D.sub.s](Q)       Discounted cost of placing an order of size
                   [Q.sub.s] for s = 1,2
[U.sub.s](Q)       Undiscounted cost of placing an order of size
                   [Q.sub.s] for s = 1,2

2. Elements of the temporary-price-reduction problem, TPRP

The first variation, in the length of the grace period, delimits the time within which the special order(s) may be placed. Such period starts at [t.sub.B] and ends at [t.sub.F]. The issues here are: (i) whether or not the regular ordering points associated with a no-discount pricing policy correspond to the end of the grace period; and (ii) the number, if any, of regular reorder points that may fall within the grace period. The end results are four cases, characterized as follows.

Property 1: grace periods vs. regular ordering points

Case 1: [t.sub.2] [is less than] [t.sub.B] [is less than] [t.sub.F] = [t.sub.1]

The grace period falls within two consecutive regular reorder points with the end of the replenishment cycle coinciding with the end of the grace period.

Case 2: [t.sub.2] [is less than] [t.sub.B] [is less than] [t.sub.F] [is less than] [t.sub.1]

The grace period falls strictly within two consecutive regular reordering points, with the end of the replenishment cycle not coinciding with the end of the grace period.

Case 3. [t.sub.2] [is less than] [t.sub.B] [is less than] [t.sub.1] [is less than] [t.sub.F]

The grace period includes only one regular reordering point, which does not coincide with the end of the grace period. Case 4. [t.sub.B] [is less than] [t.sub.r] [is less than] ... [is less than] [t.sub.1] [is less than or equal to] [t.sub.F]

The grace period includes r [is greater than] 1 regular reordering points.

The other two variations relate the minimum purchase required to qualify for the discount to the exact amount actually qualifying for the discount. This comparison gives rise to the two scenarios described in the next property.

Property 2: minimum purchase vs. discounted order size

Scenario 1

Discount is given only on amounts exceeding Y units, as long as the order size is at least X [is greater than or equal to] Y [is greater than or equal to] 0 units.

Scenario 2

Discount is given only on amounts not exceeding Y units, as long as the order size is at least X [is greater than or equal to] Y [is greater than or equal to] 0 units.

Simplified versions of Scenario 1 have been studied elsewhere. The X = Y = 0 case has been discussed in Arcelus and Srinivasan (1992), Ardalan (1988), Aull-Hyde (1992), Baker (1976), Tersine and Gengler (1982) and Tersine and Price (1981). Models with Y = 0 and X = [Q.sub.0] appear in Arcelus and Srinivasan (1992) and Ardalan (1988), with X [is greater than] 0 and Y = 0 in Aull-Hyde (1992), and with X = Y = [Q.sub.0] in Arcelus and Srinivasan (1992). The generalization of this paper removes the constraints on the values of X and Y, beyond the necessary condition X [is greater than or equal to] Y [is greater than or equal to] 0. Scenario 2 introduces a rationing policy, which limits the discount to a maximum of Y units per order. Such anti-hoarding practices, so common in both retailing and manufacturing, have not been incorporated into any of the existing TPRP models.

3. Evaluation of strategies

Combining the four cases of Property 1 with the two scenarios of Property 2 yields eight different versions of the TPRP for which solution strategies are provided below. For each version the evaluation process consists of the following elements:

(a) The planning horizon needed for the evaluation of the strategies. Given the short-term and temporary nature of the TPRP, a relatively small planning horizon is needed. For convenience, the yearly demand, R, is used in all cases.

(b) The decision alternatives. All versions have a common alternative, the status quo, which implies not deviating from the regular order schedule in the presence of the price discount offer. Any other is determined on the basis of the specific case under consideration. For versions that include Cases 1 or 2, a second alternative is possible, namely placing an extraordinary order at [t.sub.F]. For Case 3 there is the additional issue of what to do with the demand q = R([t.sub.F] - [t.sub.1]), between [t.sub.1] and [t.sub.F]. The question of interest is whether a single extraordinary purchase is to be placed at [t.sub.1] or a combined partial order of q units at [t.sub.1] and a special order at [t.sub.F] is preferable. Finally Case 4's versions add another set of problems to consider. At issue here are the size and magnitude of the order(s) during the grace period. This includes (i) when and how many units ought to be purchased at the last replenishment point of the grace period, which may or may not coincide with [t.sub.F]; and (ii) the number and size of the orders needed to satisfy the demand from t, to the last reordering point of the grace period.

(c) The cost functions associated with the eight versions of the TPRP. All cost functions consist of three sets of cost components, i.e. (i) the cost of the last order before the end of the grace period; (ii) the cost of the purchasing policy before the last order; and (iii) the cost of satisfying the demand from the end of the last ordering cycle to the end of the planning horizon. For the computation of the last set, the assumption of Naddor (1966) is used, namely that the inventory cost for a fraction of a year equals that fraction of the yearly costs. Although this procedure yields only an approximation, it has been found (Yanasse, 1990) that such average cost assumption does not generally lead to significant errors, because the resulting optimal ordering policies minimize both the average cost and the deviation from the true cost. Hence, Naddor's assumption permits us to determine the cost of the orders placed after [t.sub.F] as a fraction of the yearly cost. Let [Q.sub.E] be the demand already met with the purchases made at or immediately before [t.sub.F]. Then the cost of satisfying the remaining units, R - [Q.sub.E], may be expressed, for either scenario, as

(1) E([Q.sub.E]) = (R - [Q.sub.E])(PR + PF[Q.sub.0])/R

Finally, each set of costs is composed of purchasing, ordering and holding costs. For the computation of the latter it should be observed that the amount in stock, as well as that used to satisfy demand, can be of two prices, thus bringing to the fore the question of inventory valuation. However, all units in a given order are assumed to have arrived simultaneously, regardless of price, and are then sold uniformly throughout the replenishment cycle. Thus it is impossible to separate, as the inventory is depleted, the cheaper from the most expensive units. Hence, neither LIFO nor FIFO, with their emphasis on the identification of the physical flows, are applicable to this situation. Accordingly the other most common valuation approach, namely the average cost method, is used in this paper. Consequently, the cost of holding inventory is computed as the weighted average of the costs of holding the discounted and the undiscounted components of the order, with the weights being the relative portions qualifying or not for the discount. Mathematically, for an order of size [Q.sub.h], the holding cost, [H.sub.s]([Q.sub.h]), during the [Q.sub.cs]/R period may be written as

(2) [H.sub.s]([Q.sub.h]) = [Q.sub.h](P'F[Q.sub.h] + dFY)/2R for s = 1, = [Q.sub.h](PF[Q.sub.h] - dFY)/2R for s = 2,

and the total cost for a discounted, [D.sub.s]([Q.sub.h]), and an undiscounted [U.sub.s]([Q.sub.h]) order, respectively, as

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(d) The optimal policy for each version, which corresponds to the alternative that yields the number, magnitude and timing of the least-cost purchasing strategy.

Case 1: [t.sub.2] [is less than] [t.sub.B] [is less than] [t.sub.F] = [t.sub.1]

Let [Q.sub.1s], s = 1,2, be the size of the special order to be placed at [t.sub.F]. Then Case 1 may be described by the following property.

Property 3: characteristics of Case 1, Scenarios 1 and 2 (3a) Alternatives

Strategy A:

Buy [Q.sub.1s] [is greater than or equal to] X, s = 1,2, at [t.sub.F],

Y units for P and [Q.sub.11] - Y for P', if s = 1,

Y units for P' and [Q.sub.12] - Y for P, if s = 2; resume regular ordering of [Q.sub.0] units per order afterwards.

Strategy B:

Continue with the regular ordering policy of [Q.sub.0] units per order at the normal replenishment point. (3b) Cost functions

For s = 1,2,

[TC.sub.A] = [D.sub.s]([Q.sub.1s]) + E([Q.sub.1s]), [TC.sub.B] = U([Q.sub.0])R/[Q.sub.0] = PR + PF[Q.sub.0] if [Q.sub.0] [is less than] X, = [D.sub.s]([Q.sub.0]) + E([Q.sub.0]) if [Q.sub.0] [is greater than or equal to] X.

(3c) The optimal special orders for Case 1 are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3d) The optimal ordering strategies may be summarized as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

select B otherwise.

Case 2: [t.sub.2] [is less than] [t.sub.B] [is less than] [t.sub.F] [is less than] [t.sub.1].

The development of the optimal strategies for Case 2 proceeds along the same lines as for Case 1. Let [Q.sub.2s], s = 1,2, be the size of the extra purchase. Property 4 describes the main elements of this case.

Property 4: characteristics of Case 2, Scenarios 1 and 2 (4a) Alternatives

Strategy A:

Buy [Q.sub.2s] [is greater than or equal to] X, s = 1,2, at [t.sub.F],

Y units for P and [Q.sub.21] - Y for P', if s = 1,

Y units for P' and [Q.sub.22] - Y for P, if s = 2; resume regular ordering of [Q.sub.0] units per order afterwards.

Strategy B:

Continue with the regular ordering policy of [Q.sub.0] units per order at the normal replenishment point. (4b) Cost functions

For s = 1,2, [TC.sub.A] = [D.sub.s]([Q.sub.2s]) + 2q[H.sub.s]([Q.sub.2s]) + E([Q.sub.2s]), [TC.sub.B] = [U.sub.s]([Q.sub.0)R/[Q.sub.0] = PR + PF[Q.sub.0].

(4c) The optimal special orders for Case 2 are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(4d) The optimal ordering strategies are summarized as follows:

Select A if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

select B otherwise.

Case 3. [t.sub.2] [is less than] [t.sub.B] [is less than] [t.sub.1] [is less than] [t.sub.F].

For Case 3, the main issue is when to place the extra-ordinary order. At [t.sub.1] a replenishment is required, because the inventory level is zero. Its size may be (i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], s = 1,2, to be depleted after [t.sub.F]; or (ii) a smaller amount, [[bar]q.sub.s], s = 1,2, (shown to be equal to q), followed by another extra purchase of [[bar]Q.sub.s], s = 1,2, units (shown to be equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at ([t.sub.1] + [[bar]q.sub.s]/R) [is less than or equal to] [t.sub.F]; or (iii) the status quo quantity, [Q.sub.0], if the decision is not to deviate from the regular ordering schedule. Formally, the characteristics of Case 3 are stated in Property 5 below.

Property 5: characteristics of Case 3, Scenarios 1 and 2 (5a) Alternatives

Strategy A:

Buy [Q.sub.3s] [is greater than or equal to] X, s = 1,2, at [t.sub.1],

Y units for P and [Q.sub.31] - Y for P', if s = 1,

Y units for P' and [Q.sub.32] - Y for P, if s = 2; resume regular ordering of [Q.sub.0] units per order afterwards.

Strategy B:

Buy [[bar]q.sub.s] [is less than or equal to] q, s = 1,2, units at [t.sub.1]; buy [[bar]Q.sub.s] [is greater than or equal to] X, s = 1,2, units at [t.sub.qs] = ([t.sub.1] + [[bar]q.sub.s]/R) [is less than or equal to] [t.sub.F]; resume regular ordering of [Q.sub.0] units per order afterwards.

Strategy C:

Continue with the regular ordering policy of [Q.sub.0] units per order at the normal replenishment point. (5b) Cost functions

For s = 1,2,

[TC.sub.A] = [D.sub.s]([Q.sub.3s]) + E([Q3.sub.s]),

[TC.sub.B1] = [U.sub.s]([[bar]q.sub.s]) + [D.sub.s]([[bar]Q.sub.s]) + E([[bar]q.sub.s] + [[bar]Q.sub.s]), [[bar]q.sub.s] [is less than] X,

[TC.sub.B2] = [D.sub.s]([[bar]q.sub.s]) + [D.sub.s]([[bar]Q.sub.s]) + E([[bar]q.sub.s] + [[bar]Q.sub.s]), [[bar]q.sub.s] [is greater than or equal to] X,

[TC.sub.C1] = [U.sub.s](R) = PR + [PFQ.sub.0], [Q.sub.0[is less than] X,

[TC.sub.C2] = [D.sub.s]([Q.sub.0]) + E([Q.sub.0]), [Q.sub.0] [is greater than or equal to] X,

where B1 (B2) denotes strategy B for [[bar]q.sub.s] [is less than] X ([[bar]q.sub.s] [is greater than or equal to] X), and C1 (C2) denotes strategy C for [Q.sub.0] [is less than] X ([Q.sub.0] [is greater than or equal to] X). (5c) The optimal special orders for Case 3 are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(5d) The optimal ordering strategies are summarized in Table 2.

[TABULAR 2 DATA NOT REPRODUCIBLE IN ASCII]

The cost functions in (5b) for alternatives B and C have been broken down into two parts to reflect the undiscounted ([TC.sub.B1], [TC.sub.C1]) and discounted ([TC.sub.B2], [TC.sub.C2]) cases. The differences capture the savings in purchasing and holding costs associated with the more generous discount schedules. To derive the optimal special orders of (5c), it can be readily seen that (i) the optimal order strategy for alternative A, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], equals that for the first case; and (ii) [TC.sub.B1] and [TC.sub.B2] are decreasing functions of [[bar]q.sub.s], s = 1,2. Hence, the optimal partial order must be as large as possible and thus equal to the demand, q, for the ([t.sub.1], [t.sub.F]) period. This rules out the possibility of placing various orders between [t.sub.1] and [t.sub.F] and it also delays the placing of the larger purchase as much as possible. Then at [t.sub.F], the decision is similar to that for Case 1, alternative A, which leads to the acquisition of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], units.

Case 4: [t.sub.B] [is less than] [t.sub.r] [is less than] ...[is less than] [t.sub.1] [is less than] [t.sub.F]

The distinguishing feature of Case 4 is the need to place at least one order before the extraordinary replenishment at to [is less than or equal to] [t.sub.F]. Hence, the planning horizon may be divided into three periods. This includes the two listed in Case 3, plus the interval covering the ([t.sub.r], [t.sub.0]) period. For the latter, the main issues to be resolved are (i) the number of orders required to satisfy the R([t.sub.0] - [t.sub.r]) demand and (ii) the feasibility of its corresponding order size, with respect to X and to the demand, [Q.sub.f], during the ([t.sub.r], [t.sub.F]) interval. Property 6 includes the main features of this case.

Property 6. characteristics of Case 4, Scenarios 1 and 2. (6a) Alternatives

For each strategy a = A,B and each scenario s = 1,2,

1. Buy [g.sub.as] orders of size [[bar]Q.sub.as], starting at [t.sub.r], where

[g.sub.as][[bar]Q.sub.as] [is less than or equal to] [Q.sub.f] [is less than or equal to] ([g.sub.as] + 1)[[bar]Q.sub.as],

[[bar]Q.sub.as] [is greater than or equal to] x, if strategy a = A, [is less than] X, if strategy a = B.

2. Buy [Q.sub.1s] units at time [t.sub.0] = [t.sub.r] + [g.sub.as][[bar]Q.sub.as/R [is less than or equal to] [t.sub.F].

3. Resume regular ordering of [Q.sub.0] units per order afterwards.

(6b) Cost Functions.

For s = 1,2,

[TC.sub.A] = [g.sub.As][D.sub.s]([[bar]Q.sub.As]) + [D.sub.s]([Q.sub.4s]) + E([g.sub.As][[bar]Q.sub.As] + [Q.sub.4s]),

[TC.sub.B] = [g.sub.Bs] [U.sub.s]([[bar]Q.sub.Bs]) + [D.sub.s]([Q.sub.4s]) + E([g.sub.Bs][[bar]Q.sub.Bs] + [Q.sub.4s]).

(6c) The optimal ordering strategies are summarized in Table 3.

[TABULAR DATA 3 NOT REPRODUCIBLE IN ASCII]

Properties (6a) and (6b) suggest that the difference between the strategies relates only to whether or not the discount is worthwhile during the ([t.sub.r], [t.sub.0]) time period. The tradeoff is between the lower ordering and purchasing costs of the larger order size, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], of strategy A, against the lower holding costs of strategy B. The end result is (i) a uniform purchasing strategy, A, where the order size is that of Case 1; and (ii) another, B, that leads to the regular ordering schedule in the presence of the price discount offer. The dominance conditions for the strategies are listed in Table 3. Proofs are given in Appendix A. Because all share the common post-[t.sub.0] policy listed in (6a), each alternative is defined only in terms of the pre-[t.sub.0] values of the order size and of the number of orders. For the undiscounted case, the policy is clear in either scenario. If feasible, the [Q.sub.0] is the best option for strategy B. Otherwise, A is preferable, because a discounted [Q.sub.0] will always yield a lower-cost policy than any undiscounted quantity. Obviously, in the most common situation of X = 0, B is unfeasible. Only when the attractiveness of A diminishes as the size of the order required for the discount increases does the status quo become a more acceptable option. Within strategy A, note that the policy with the optimal order size, [Q.sub.1s] is not necessarily the best, because the number of orders must also be considered as a decision variable. Fortunately, as Appendix A shows, at most one other order size requires evaluation. Once again, the difference between the scenarios is manifested clearly in their respective optimal policies. The more liberal discount policies of Scenario 1 result in a higher likelihood that the ([m.sub.1], [Q.sub.11]) policy dominate, to the point where outright dominance occurs under the first two sets of conditions listed in Table 3.

4. Conclusions

This paper has endeavoured to generalize the temporary price reduction problem, TPRP, through the four-case, two-scenario framework presented in the preceding sections. This has resulted in an increase in the applicability of the TPRP, through the incorporation of a variety of new practical concerns. Another contribution of the generalized TPRP lies in its ability to identify the type of discounting practices that run with or against current purchasing practices. In this way, it is possible to determine which situations lead to the type of small-order-size policies favoured today and which contradict other inventory-cutting practices. In addition, the model also serves to illustrate that discounting policies do not necessarily have to result in larger inventory levels.

Acknowledgements

Financial support from the Natural Sciences and Engineering Research Council of Canada for the completion of this research is gratefully acknowledged. The authors would also like to thank the referees for their comments on earlier versions of the paper. Special thanks go to the referee who suggested the change in Scenario 1 from Y [is less than or equal to] X [is less than or equal to] [Q.sub.0] to 0 [is less than or equal to] Y [is less than or equal to] X

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Appendix A. Derivation of the optimal strategies for Case 4

Because each Case 4 policy is characterized by two pre-[t.sub.0] parameters, the number of orders to be placed and their size, so are the strategies. Hence, both [g.sub.as] and [bar][Q.sub.as] must be derived for each scenario and for both the discounted (a = A) and the undiscounted strategies (a = B).

Discounted Strategy

Standard optimality conditions lead to the following lemma.

Lemma 1. The policy associated with the lowest cost order quantity is characterized by [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]], where ms, is the largest integer not exceeding [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and

(A1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The policy defined in Lemma 1 is not necessarily optimal, as there are others that may also satisfy (A1). Denote these policies as [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]] where ([n.sub.s]-[m.sub.s]) and [[Delta].sub.s] are either both positive or both negative. The following lemma establishes the best order quantity for each [n.sub.s].

Lemma 2. For each order size [n.sub.s], the lowest-cost order quantity satisfying (A1) is [Q.sub.f]/[n.sub.s].

Proof: From Property (6b), it can he readily seen (i) that the cost differentials between [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]] and [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]] are

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and (ii) that the first derivative of [B.sub.s] with respect to [[Delta].sub.s], [B'.sub.s]([[Delta].sub.s]), is equal to

(A3) [B'.sub.s]([[Delta].sub.s]) = P'[Fn.sub.1] [[Delta].sub.1]/R for s = 1, P'[Fn.sub.2] [[Delta].sub.2]/R for s = 2.

From (A3), it is clear that [B'.sub.s]([[Delta].sub.s]) is positive when [[Delta].sub.s] [is greater than] 0 and negative when [[Delta].sub.s] [is less than] 0. Hence, the largest [B.sub.s] corresponds not only to the smallest [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] but also to the largest order quantity satisfying (A1), i.e., to [Q.sub.f]/[n.sub.s].

The next step is to devise a stopping rule, which determines the number of order sizes, [n.sub.s], to compare with the [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] policy. The answer to this query is provided by the next two lemmas, which show that at most one [[n.sub.s], [Q.sub.f]/[n.sub.s]] policy for each scenario is required.

Lemma 3.

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof: The lemma is a direct outgrowth of (A2). For s = 2, it should also be observed that the definition of [Q.sub.0], along with Property (3c), yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 3 justifies the dominance relations of Table 3. When the conditions of (A4) are not satisfied the next lemma establishes the existence of a unique [n.sub.s].

Lemma 4. Only the [[n.sub.s], [Q.sub.f]/[n.sub.s]] policy defined in Table 3 needs to be compared with [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]].

Proof: Standard optimality conditions indicate that the least-cost order quantities sizes, assuming real rather than integer [n.sub.s], may be written as

(A5) [Q.sub.f]/n5(real) = max[X, [square root of 2R(K + dY)/PF]], for s = 1, = max[X, [square root of 2R(K - dY)/PF]], for s = 2.

Further, the concavity of the cost functions suggests that for integer order quantities the minimum occurs at the value that satisfies the following condition:

(A6) [TC.sub.s][([n.sub.s] - 1), [Q.sub.f]/([n.sub.s] - 1)] [is greater than or equal to] [TC.sub.s] [[n.sub.s], [Q.sub.f]/[n.sub.s]] [is less than or equal to] [TC.sub.s] [([n.sub.s] + 1), [Q.sub.f]/([n.sub.s] + 1)].

Simple algebraic manipulation with (A6) leads to the expressions in Table 3.

Undiscounted Strategy

Following an approach similar to that for the discounted strategy, the following two lemmas establish (i) the existence of a lowest-cost order quantity, [Q.sub.0]; and (ii) the dominance of the resulting [[m.sub.0], [Q.sub.0]] policy.

Lemma 5. For either scenario, the policy associated with the lowest-cost undiscounted order quantity is characterized by [[m.sub.0], [Q.sub.0]], where [m.sub.0] is the largest integer not exceeding [Q.sub.f]/[Q.sub.0].

Lemma 6. Undiscounted [[m.sub.0], [Q.sub.0]] policy dominates any other undiscounted policy for either scenario.

Proof. The cost differential between [[m.sub.0], [Q.sub.0]] and any other undiscounted [[n.sub.s], [Q.sub.0] - [[Delta].sub.s]] may be expressed as

(A7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is always negative.

Finally, the strategy in Table 3 is justified by observing that, if [Q.sub.0] [is greater than or equal to] X, any other undiscounted policy is worse than the undiscounted [Q.sub.0] policy, which in turn is dominated by the non-optimal discounted [Q.sub.0] policy. Under these conditions, no undiscounted policy is worth considering.

Biographies

F.J. Arcelus is a Professor of Quantitative Methods in Management at the University of New Brunswick, Canada. He received a B.Sc. in Quantitative Methods in Business and Economics from The California State University at Northridge and an MS in Industrial Administration and a Ph.D. in Urban and Public Affairs from Carnegie-Mellon University. He is currently involved an interdisciplinary research in the areas of Accounting Policy, Regional Planning and International Business, in addition to his major research interests in Inventory Control and Production Planning. He has published articles in a variety of journals, including Management Science, Technometrics, IIE Transactions, European Journal of Operations Research, Decision Sciences, International Journal of Productions Research, Journal of the Operational Research Society, International Journal of Production Economics, Accounting Horizons, Public Finance, Mathematical Social Sciences, Journal of Money, Credit and Banking and Growth and Change.

G. Srinivasan is a Professor of Finance at the University of New Brunswick, Canada. He received a Masters degree in Commerce from Sri Ventkateswara University and a doctoral degree from the Indian Institute of Management at Ahmedabad, India, as well as the professional accounting designation of CGA Canada. He is currently doing research in the areas of Inventory Models, International Business, Fiscal Incentives and Investment and Accounting Policies. His papers have appeared in a number of journals including Management Science, Shankya, European Journal of Operations Research, Decision Sciences, Journal of the Operational Research Society, International Journal of Production Economics, Omega and International Journal of Management.

Received September 1991 and accepted August 1993