Omega 29 (2001) 417–428
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Retailer’s response to special sales: price discount vs. trade credit F.J. Arcelusa; ∗ , Nita H. Shahb , G. Srinivasana a Faculty
of Administration, University of New Brunswick, PO Box 4400, Fredericton, N.B. Canada E3B 5A3 of Mathematics and Statistics, Gujarat University, Ahmedabad, 380009, Gujarat, India
b Department
Received 7 May 1999; accepted 15 May 2001
Abstract Given the increasing saliency of special o1ers as a sales promotion tool, this paper analyses the advantages and disadvantages of the two most common payment reduction schemes, namely a decrease in the purchase price and a delay in the payment of the merchandise. Following some of the latest empirical evidence in the sales promotion 2eld, the model includes a price-dependent demand, where price incorporates the ability of the retailer to pass on some of the savings to the customers. The integration of both the purchasing and the sale implications of the vendor’s o1er on the retailer’s pro2t forms an integral part of the model. A numerical example highlights the main features of the model. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Inventory; Purchasing; Pricing; Deterioration; Forward buying
1. Introduction The practice of vendors o1ering special incentives to retailers for a limited time period to o;oad excess inventory is quite prevalent in some industries today [1,2]. In fact, these practices are even beginning to take precedence over advertising in the sales promotion budget of many 2rms [3,4] in spite of the on-going controversies over their pro2tability [5 –8]. Quite a few examples of these practices have been documented in the literature for several sectors of the economy e.g. [9 –11,1], including pharmaceuticals, hardware, furniture, fashion apparel, and frozen food. These vendor-to-retailer incentives, denoted by trade promotions in the sales-promotion literature [3], can take on many forms, with discounts on the purchase price (DPP) and a delay of payment (DOP) being the most prevalent. It is the
∗ Corresponding author. Tel. +1-506-453-4869; fax: +1-506-453-3561. E-mail addresses:
[email protected] (F.J. Arcelus), nita sha h@ hotmail.com (N.H. Shah),
[email protected] (G. Srinivasan).
purpose of this paper to develop the retailer’s pro2t maximizing strategy in answer to these promotions. By their very nature, these incentives will impact on the retailer’s cost and revenue structures. On the cost side, these e1ects are manifested through modi2cations in both the ordering and the inventory policies, as the retailer places a special order from the vendor in response to the trade promotion under consideration. In turn, the resulting cost decreases leave open the possibility for the retailer to design its own retail promotion, by reducing the price it charges to its own customers for the merchandise, with the corresponding expected rise in the demand for the merchandise and hence on the resulting sales revenues. As a result, the paper models the retailer’s decision process through the design of the pro2t-maximizing ordering and retail promotion strategies in response to the vendor’s trade promotion. The integration of both the purchasing and the sale implications of the vendor’s o1er on the retailer’s pro2t forms an integral part of the model. There exists an extensive literature both in operations management and in marketing, concerning the retailer’s strategic decision, when confronted with a DPP special
0305-0483/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 4 8 3 ( 0 1 ) 0 0 0 3 5 - 4
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sale. On the operations management side, the basic formulation of Baker [12] appears in many textbooks on the subject, such as in Tersine [13]. Many other studies have attempted to adapt Baker’s model [12] to handle model concerns. Examples of this work include Ardalan [14], to incorporate discount periods not coinciding with the replenishment cycle and Arcelus and Srinivasan [15], to consider cases where the retailer’s special purchase is large enough to encompass several no-discount ordering cycles. Most of this extensive operations management literature has been reviewed elsewhere [16,17]. A common assumption of the majority of these studies is that of a constant demand, independent of sales-price Juctuations. Hence, the buyer=retailer’s ability to inJuence demand by passing on at least some of the discount to its customers has been ignored, with the exceptions of those studies [18,16,19] which incorporate the price=demand relation. An important contribution of Ardalan [19] is to show that the usual formulation of the constant demand case, designed to decide whether the special o1er should be accepted or rejected, is no longer relevant when the demand is price dependent, since the acceptance option is always dominant. However, the models use the limited strategy of passing on the entire discount on all the units bought in the special sale, even though the marketing evidence clearly indicates e.g. [20 –23,3] that this is not necessarily the case. With respect to the shape of the demand function, the operations management literature favours the linear demand, as a convenient two-term Taylor approximation to the more complex and more theoretically appealing exponential counterpart [24], with its constant price elasticity of demand. The marketing literature has gone much further in the characterization of DPP behaviour, specially as it refers to the implications of the functional demand form, be it linear or non-linear, on channel behaviour e.g. [6,8,22,23,25]. Nevertheless, it is also common practice e.g. [4], for wholesalers to resort to the alternate strategy of allowing credit, instead of discount, to induce larger orders, especially when the fear of competitive reaction to a DPP policy is high [26 –29]. However, the fact that a DOP policy conveys a new bene2t to the retailer, for whom it may be bene2cial to pass on at least a portion of it to its customers to induce a higher demand level, has not yet been addressed. In an attempt to bridge this gap, the present study considers both the DPP and the DOP options for a pro2t maximizing retailer with a price-dependent demand. The basic model for both options is presented in the next section, together with the derivation of the optimality conditions. In accordance with the empirical evidence alluded to earlier, the model considers the optimal determination of the amount of discount to be passed on to the customers and the quantity eligible for such a discount. This is followed in Section 3 by an analytical and numerical comparison of the resulting policies. Conclusions given in Section 5 completes the paper.
2. Components of the buyer’s decision models for the two options The 2rst step in the modelling process is to determine the length of the planning horizon to be used for comparability between the two options. Even if the inventory problem under consideration is an in2nite horizon problem, all that is required for the evaluation of DOP and DPP is a period long enough to cover the depletion periods of the special orders placed by the retailer in response to the two options. Since it is not clear a priori which depletion period is larger, a one-year planning horizon is selected and the time-dependent parameters and variables are de2ned accordingly. However, should at least one of the depletion periods exceed one year in length, as it will be for some portions of the analysis in Section 4, a longer one may be selected without adding complexity to the model. The potential time di1erence between the end of the depletion period and that of the planning horizon indicates the need for the calculation of the buyer’s pro2t for the no-incentive in-between period. Following the standard assumption of Naddor [30], its value corresponds to the fraction of the yearly no-incentive pro2t represented by the length of the in-between period. This necessitates the derivation of a pro2t function for the no-incentive situation to go along with the those corresponding to the DPP and the DOP options. An important ingredient of this derivation is the modelling of the price-dependent demand, denoted by Rj , j = 0; 1; 2 (j = 0 represents the no-incentive case) and de2ned by the standard downward-sloping price-dependent demand, as follows: Rj = R(Pj )
with @Rj =@Pj = Rj ¡ 0;
@2 Rj =@2 Pj = Rj unrestricted in sign
for j = 0; 1; 2:
(1)
The economics and operations management literature adds another restriction to Rj , namely that of concavity, which requires Rj ¿ 0. With this condition added to (1), demand is assumed to decrease at an increasing rate as price rises. Tyagi [23] considers also convex demand functions, which under certain conditions, may yield greater than 100% retail pass-through behaviour. That Rj in (1) is unrestricted in sign indicates that the models of this paper can accommodate both, convex and concave demand functions. Table 1 lists the various elements of the pro2t functions. Pro2ts may be decomposed into two components. One, associated with the 2rst cycle, covers the special-order period of length as listed in Table 1. The other covers the rest of the planning horizon. It should be emphasized once again that the no-incentive policy is not an option per se, but, as shown below, it forms part of the other two and has a similar cost=revenue structure. However, for comparability purposes to the DPP and DOP options, its 2rst cycle, of equal length, Q0 =R0 , to all other cycles, is also described in detail. Hence, for the no-incentive policy, j = 0, the revenue for the special cycle is generated by the sale of the 2rst Q0 units at the regular price, P0 . The functional form of the
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Table 1 Modelling the options No discount B0 a
Discounted purchase price (DPP) B1
Delay of payment (DOP) B2
Length of the 2rst cycle
Q0 =R0
(Q1 − q1 )=R1 + q1 =R0
(Q2 − q2 )=R2 + q2 =R0
+ Revenues
P 0 Q0
P1 (Q1 − q1 ) + P0 q1
P2 (Q2 − q2 ) + P0 q2
Acquisition costs
K + cQ0
K + (c − d)Q1
K + cQ2
Physical holding costs
k[(Q12
k[(Q22 − q22 )=R2 + q22 =R0 ]=2
Financial holding costs
kQ02 =2R0 cFQ02 =2R0
+ Pro2t for the rest of the year
(1 − Q0 =R0 )B0
B0 {1 − [(Q1 − q1 )=R1 + q1 =R0 ]}
Total pro2t = + Pro2t from the 2rst cycle =
− Costs
aB
0
(c −
− q12 )=R1 + q12 =R0 ]=2 d)F[(Q12 − q12 )=R1 + q12 =R0 ]=2
cF[(Q22 − q22 )=R2 + q22 =R0 ]=2 − cFtQ2 B0 {1 − [(Q2 − q2 )=R2 + q2 =R0 ]}
= R0 (P0 − c) − KR0 =Q0 − (k + cF)Q0 =2
special-order revenues for options j = 1(DPP); 2(DOP) allows the buyer to sell (Qj − qj ) units for a promotional time period of length j = (Qj − qj )=Rj at the discounted price of Pj and for a (qj =R0 ) period, the remaining qj units at the regular no-discount price of P0 . As a result, the depletion period of the discounted Qj units, denoted by Tj , is given by Tj = j + qj =R0 ;
j = 1(DPP); 2(DOP)
= (Qj − qj )=Rj + qj =R0 :
(2)
The cost expressions start with the acquisition costs associated with the special (for j = 1; 2) or the no-incentive (for j = 0) purchasing policies. All include a 2xed component, K, per order, plus the variable cost, at a lower rate for the DPP option than for the other two, of purchasing the special order of size Qj , j = 0; 1; 2. Holding costs are of two types and are incurred regardless of the discount option under consideration. There are the physical costs of handling the merchandise, such as insurance, storage and the like. These charges normally take the form of a per-unit charge (k$=unit), on the basis of the average inventory on hand throughout the cycle. Then, there are the 2nancial costs of holding the merchandise. These costs are of the order of cF per unit per year for the no-discount cases (j = 0; 2) and of (c − d)F for the DPP (j = 1) option and are also usually charged on the basis of the average inventory on hand throughout the cycle. As shown in Table 1, both the physical and the 2nancial costs have similar functional structures. For j = 0, the average number of units is Q0 =2 for a period of length Q0 =R0 . For the other two, (i) the 2rst cycle of length Tj must be divided into two sub-periods of length j = (Qj − qj )=Rj , and qj =R0 , respectively, j = 1; 2; (ii) the two demand rates reJect the di1erent selling prices, Pj or P0 , being charged for the merchandise during each sub-period; and (iii) for each option j = 1; 2, the beginning and ending inventories are, Qj and qj , for the promotional period of length j . In addition to having di1erent per-unit charges, the distinguishing feature between the two options is that
the delay on the payment of the special purchase of size Qj in the DOP case (j = 2), for a time period of length t, results in a corresponding decrease, proportional to the length of the delay period, in the 2nancial cost, cFtQ2 , of handling the special sale under such option. In fact, it can be readily seen [26] that this non-linearity in the e1ect of a change in the credit period on pro2tability results in the absence of a one-to-one linear correspondence between the optimal policies derived from the DPP and the DOP options. The last component of each pro2t function is the net pro2t earned in the period between the end of the 2rst cycle and the end of the planning horizon. The expressions of the next to the last row in Table 1 reJect the use of Naddor’s assumption [30], whereby the net pro2t for the rest of the year is computed as a fraction of B0 equivalent to the proportion of time left in the planning horizon at the end of the incentive period. Finally, the di1erence between the various revenues and costs yield the appropriate total pro2t of Bj , j = 0; 1; 2. In addition, it should be observed that the value of B0 which appears in the expressions for B1 and B2 corresponds to the optimal pro2t for the no-incentive policy. Such value is a parameter to the models of this paper, as it corresponds to predetermined policy of a pro2t-maximizing buyer. Nevertheless, the optimality conditions for this case are provided in the next section, along with those for the two options.
3. Derivation and comparison of the optimal policies On the basis of the pro2t functions summarized in Table 1, the optimizing problems for j = 0; 1; 2 may be described as follows. The decision variables are (P0 ; Q0 ) for j = 0 and (Pj ; Qj ; qj ) for j = 1; 2. The objective in each case is to 2nd the values of the appropriate decision variables so as to maximize Bj , j = 0; 1; 2. The usual asterisk (∗ ) next to a variable is used to denote optimality. The optimal policies are derived in the usual way, by setting the 2rst derivatives of Bj , j = 0; 1; 2, equal to zero and solving for the respective
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F.J. Arcelus et al. / Omega 29 (2001) 417–428 Table 2 Optimality conditions No-discount case Q0∗ = 2KR∗0 =(cF + k) kQ0∗ = R∗0 + (P0∗ − c)R∗0 Discounted purchase price (DPP, j = 1) and Delay-of-payment (DOP, j = 2) options Qj∗ = [R∗j (Pj∗ − cj + sj ) − B0∗ ]=[cj F + k] qj∗ = [(P0∗ − Pj∗ )=(1=R∗0 − 1=R∗j ) − B0∗ ]=(cj F + k) 2
−R∗j =R∗j = B0∗ + (cj F + k)(Qj∗ + qj∗ )=2 where (cj ; sj ) = (c − d; 0) if j = 1
= (c; cFt) if j = 2 Main diagonal elements of the Hessian matrix @2 Bj =@Qj2 = −2KR0 =Q03 ¡ 0
if j = 0
= −(k + cj F)=Rj ¡ 0 if j = 1; 2 @2 Bj =@qj2 = − (k + cj F)(1=R0 − 1=R1 ) ¡ 0 if j = 1; 2 @2 Bj =@Pj2 = X0 = 2R0 + (P0 − c − kQ0 )R 0 ¡0
if j = 0 and X0 ¡ 0
= Xj (Qj − qj )[H + (cj F + k)(Qj + qj )=2]=R3j ¡ 0 2 for j = 1; 2 where Xj = Rj R j − 2Rj
decision variables. Table 2 lists (i) the systems of equations resulting from setting equal to zero the partial derivatives of each pro2t with respect to their corresponding decision variables; and (ii) the main diagonal elements of the Hessian matrix. The key implication of these results is that they establish necessary conditions for Bj , j = 0; 1; 2, to have no minimum. This justi2es the following property. Property 1 (P1). Bj is a partially concave function (i) of Qj , for j = 0; 1; 2; (ii) of qj , for j = 1; 2; but (iii) of Pj , only if Xj ¡ 0, for j = 0; 1; 2. That a function is “partially concave” with respect to a given variable implies that the function’s second derivative with respect to the variable in question is negative. Hence, if a maximum exists, then it is provided by the conditions of Table 2. Further, partial concavity can be readily shown to occur for the linear demand case to be used in the next section, since then the Xj ¡ 0 condition always holds for that particular functional form. The end result is that, even if the existence of a unique feasible maximum cannot be guaranteed, P1 implies that, if one is found numerically as a solution of the optimality conditions of Table 2, then it is unique. Hence, the numerical solutions show the feasibility and existence of a maximum and P1 establishes its uniqueness. Nevertheless, in our extensive computational experience, we have failed to encounter a case where no feasible solution exists. After adroit manipulation of the analytical expressions, further examination of Table 2 suggests that the optimality conditions have a very intuitively appealing marginal-revenue-equals-marginal cost economic interpretation. For both options, the 2rst line deals with the condition
if j = 1; 2 and Xj ¡ 0
on Qj , j = 1; 2. It indicates that, other things being equal, a one-unit increase in Qj causes revenues to rise by $Pj and for the DOP option, additional savings from the holding costs foregone by the delay in the payment of the extra unit of merchandise. The compensating extra costs include (i) the additional purchasing and holding costs incurred by the extra unit; and (ii) the pro2t foregone during the post-incentive period. The second line refers to the e1ects of a marginal increase in qj , j = 1; 2. Other things being equal, each extra unit sold at an incentive generates (i) a marginal revenue increase of P0 − Pj , as the extra unit is sold at the higher price; (ii) a decrease in the incentive depletion period of (1=Rj − 1=R0 ), with the appropriate changes in the holding costs and in the post-incentive period pro2ts. The third line considers the implications of a one-unit decrease in the selling price, Pj , j = 1; 2. It involves a revenue decrease from the units sold at a price discount. The counterbalance is an increase in Rj and hence in the holding costs of the incentive units, together with lower pro2ts from the post-incentive period. The next property sets a lower limit on the optimal retailer’s price discount. Its proof is given in Appendix A. Property 2 (P2). It is always pro:table for the retailer to pass on at least some of the vendor’s incentive to its own customers; i.e. P0∗ − Pj∗ = ∗j ¿ 0
for j = 1; 2
(3)
P2 reJects the importance of including a price-dependent demand. A price-elastic demand ensures that at least some portion of the vendor’s incentive passes through to the ultimate consumer. In this way, the model addresses one of the key vendors issues with respect to the pro2tability of
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trade deals, namely to what extent does the retailer pass on the vendor’s incentives [3,4]. Otherwise, if demand is not price-elastic, the retailer may simply pocket the bene2ts. P2 also forms the basis for the next property.
Qj∗ ¿ Q0∗
for j = 1; 2
(4)
P3 establishes the size relationships among the order quantities. It illustrates that the retailer’s optimal policies also accomplish the vendor’s objective, namely to provide the necessary 2nancial incentives to ensure that retailers place higher order quantity than normal. The proof of P3 is also provided in Appendix A. The next property suggests that a suPcient but not necessary condition for the discount option, j = 1, to dominate its credit counterpart, j = 2, is for the value of the vendor’s credit o1er, cFt, to be less than the value of its discount o1er counterpart, d. Property 4 (P4). B 1 ¿ B2
if cFt ¡ d
(5)
P4 is only suPcient but not necessary, because, as proven in Appendix A, the j = 1 option may dominate even if d 6 cFt. In fact, indi1erence to the options, i.e. B1 = B2 , requires d ¡ cFt. Additional economic insights may be obtained by examining analytically some of the model’s parameters and their e1ect on optimality. From the pro2t and optimal conditions of Tables 1 and 2, respectively, it can be readily shown that, other things being equal, (i) increases in d, t or in the demand’s price elasticity; or (ii) decreases in c, F or k lead to higher order sizes, to lower qj ’s, to lower prices and ultimately to higher bene2ts for the retailer. 4. A numerical example This section illustrates numerically the degree to which optimal policies from both options di1er from each other in terms of pro2tability. To facilitate comparability, the example in Ardalan [19] is used in this paper, to which a physical holding charge has been added. In addition, only the linear demand case, with an intercept of r0 and a slope of r1 , is considered in this section. From results not shown here, it can be stated that the corresponding results of the relative comparison between the DOP and the DPP options for the exponential demand case are almost identical. Further, it can also be stated that the conclusions of the numerical analysis presented in this section are consistent throughout the wide variety of examples studied, but not discussed here. The basic parameter values and the corresponding optimal policies are [K c k F d r0 r1 t] = [10 10 0:5 0:25 2 49; 000 3000 0:5];
[P0∗ R∗0 Q0∗ B0∗ ] = [13:19 9440 251 29; 330]; [P1∗ R∗1 Q1∗ q1∗ B1∗ ] = [12:51 11; 480 8963 2718 38; 240]; [P2∗
Property 3 (P3).
421
R∗2
Q2∗
q2∗
(6)
B2∗ ]
= [12:76 10; 730 4554 1478 32; 260]: From (6), it can be easily seen that the DPP (DOP) option calls for placing a special order of approximately Q1 = 8963 (Q2∗ = 4554) units, at a cost of c−d = 10−2 = $8 (c = $10), per unit, payable immediately (after half, t = 0:5, a year). Of these, Q1∗ − q1∗ = 6245 (Q2∗ − q2∗ = 3076) units are sold by the retailer during the promotional period of length ∗1 = (Q1∗ −q1∗ )=R∗1 = 0:54 (∗2 = [Q2∗ −q2∗ ]=R∗2 = 0:42) of the year, at a unit price of P1∗ = $12:51 (P2∗ = $12:76), to satisfy an annual demand rate of R∗1 = 11; 480 (R∗2 = 10; 730) units. The rest, q1∗ = 2718 (q2∗ = 1478), of the units are sold at the no-discount price of P0∗ = $13:19 per unit, during the remainder of the depletion period of length q1∗ =R∗0 = 0:29 (q2∗ =R∗0 = 0:16) of the year. For the reminder of the planning horizon, 1 − 0:54 − 0:29 = 0:17 (1 − 0:42 − 0:16 = 0:42) of a year, the net pro2t amounts to 0:17B0∗ = $4986 (0:42B0∗ = $12; 318). The end result is a yearly total pro2t of B1∗ = $38; 240 (B2∗ = $32; 260). A similar analysis has been carried out for a large number of examples, as well as a sensitivity analysis for each, on the e1ect of Juctuations in the values of the parameters, within the (−50%; 50%) range, on the retailer’s optimal ordering policies for the DOP and DPP options. The results may be summarized by reporting the positive (negative) impact on pro2ts of decreases (increases) in K; k; c; F and r1 or of increases (decreases) in d and r0 . These “marginal e1ects” all fall within the usual tenets of microeconomic theory and thus represent an indirect validation of the models. These are anticipated outcomes, given the well-behaved functions in (1) and in the tables and given the deterministic environment that, even the presence of extreme parameter Juctuations, precludes deviations, commonly due to randomness, from these expected results. Four questions of interest arise from the comparison of the policies listed in (6), namely (i) the extent to which the optimal policies for each option are a1ected by their corresponding payment reduction parameter, d for DPP and t for DOP; and (ii) the extent to which the pro2t estimates of one option vary when the optimal pricing and ordering policies of the other option are used; (iii) the extent to which a feasible and practical credit period (discount level) can be derived to yield the same optimal pro2t for the retailer as that associated with a given discount level (credit period); and (iv) the extent to which there exists an opportunity loss whenever the retailer is constrained for competitive reasons to 2x the length of the promotional period, j = (Qj −qj )=Rj , j = 1; 2, within which the retailer’s price-reduction strategy may be put in operation. Figs. 1 and 2 illustrate the e1ects associated with the 2rst query for the DPP and DOP options,
422
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Fig. 1. The e1ect of changes in d on the optimality conditions.
Fig. 2. The e1ect of changes in t on the optimality conditions.
respectively. The information is presented in the form of spidergrams [31]. These graphs depict the relationships between the percentage change in d (Fig. 1) and t (Fig. 2) and the percentage change in the corresponding optimal policies (Bj∗ ; Pj∗ ; R∗j ; Qj∗ ; qj∗ , j = 1; 2) for each option. The changes are computed from the base values of (6). For example in Figs. 1 and 3, the x-axes include values of d ranging from 1 and 3, with these two values representing a 50% decrease
and increase, respectively, in the basic value of d = 2, given in (6). It should be noted that for the larger percentage increases in d, the depletion period T1 ¿ 1 and thus a two-year planning horizon is used. Another observation from Figs. 1 and 2 reJects the fact that the Juctuations are substantially more pronounced for the quantities, the Qj∗ s and qj∗ s, than for the price, to the point of observing decreasing (increasing) rates of change in prices (quantities). These observations
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423
Fig. 3. The e1ects of DPP optimal policies on the DOP option.
Fig. 4. The e1ect of DOP optimal policies on the DPP option.
are consistent with the rationale for the use of non-discount incentives alluded to earlier. If fear of retaliation from the bigger competitors is of importance, then manipulation of quantities, rather than of prices, tend to attract less attention at least in the short term. Further, the evidence on the retailer’s pro2ts, as depicted by the continuous line in each 2gure, reJects the expected increased in pro2tability, albeit at a decreasing rate, which accompany more favourable 2nancial incentives o1ered by vendors. The primary implica-
tions of these results are clear. In response to the vendor’s incentives, the retailer may alter quite pro2tably its ordering policies, through adroit manipulation of the decision variables. Retailer’s pro2tability can be further enhanced by a transfer of at least some of these bene2ts to the retailer’s own customer base through the pricing mechanism. The results of the 2gures appear to indicate that the options yield similar policies, at least on a relative basis. To further explore the extent of these similarities, Figs. 3 and 4
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F.J. Arcelus et al. / Omega 29 (2001) 417–428
Fig. 5. d and t for the same optimal pro2t.
depict the e1ect of using the optimal policies of one option as ordering and pricing policies of the other and then compute the di1erences in bene2ts with the maximum pro2ts, B1∗ and B2∗ of (6). Hence, in Fig. 3, the pro2t for the DOP option, B2∗ , with P1∗ ; Q1∗ and q1∗ , as given in (6) is compared to B2∗ , resulting from P2∗ ; Q2∗ and q2∗ , for percentage changes in t ranging from −50% to 50%. In similar fashion, Fig. 4 compares the pro2t for the DPP option, B1∗ , for P2∗ , Q2∗ and q2∗ , as given in (6), to B1∗ , resulting from P1∗ ; Q1∗ and q1∗ , for percentage changes in d ranging from −50% to 50%. The evidence from Figs. 3 and 4 illustrate clearly the di1erent direction of these e1ects. DPP pro2tability, ordering and pricing policies resemble their DOP counterparts, the larger the delay in the payment of the merchandise. As before, quantities Juctuate more widely than prices, to avoid short-term competitive reaction. The end result in terms of pro2ts, represented by the continuous lines, consists of almost-equivalent policies throughout the range of values of t utilized in the example, with percentage di1erences under 10%, for all cases. In contrast, DOP pro2tability, ordering and pricing policies resemble their DPP counterparts, the smaller the per-unit discount is. In addition, quantities Juctuate over a wider range of values than prices, also for competitive reasons. Observe also that converting the per-unit bene2t of trade credit into equivalent linear price discount will not result in identical policies. This lack of a one-to-one linear correspondence between the DOP and the DPP options can be clearly illustrated by comparing the pro2tability of the base-case DOP option given in (6), to its discount-equivalent DPP counterpart. From (6), a trade credit of t = 0:5 of a year corresponds to a bene2t of cFt = $1:25 per item and to a yearly pro2t of B2∗ = $32; 260. This amount understates the DPP
pro2t, B1∗ , associated with a discount of d = $1:25. To show this, consider Fig. 1. A value of d = $1:25 corresponds to a percentage decrease of 37.5% in the value of the base-case per-item discount (d = $2), which in turn corresponds to approximately a 10% decrease in the base-case value of B1∗ = $38; 240, for a total yearly pro2t exceeding $34,000. An alternate way of looking at the relationship between the two incentives arises from the computation of the magnitude of one of the incentives needed to obtain the same optimal pro2t to the retailer than that obtained by a given level of the other. Fig. 5 illustrates such relationship. The computational procedure is simple. Optimal discount policies are obtained for a wide range of values for d, from 0.25 to 5, representing between 2.5% and 50% discount from the base cost of c = 10, identi2ed in (6). Then, a four-equation simultaneous system is used to compute the value of the credit period, t, needed to generate an equal-pro2t optimal DOP policy. Four rather than three equations are needed since, in addition to the three DOP decision variables, namely Q2 , q2 and P2 , the discount period, t, is also a variable in this case. The system is formed by the three DOP optimality conditions of Table 2 and the corresponding DOP pro2t function from Table 1, with B2∗ being the given optimal pro2t derived from the equivalent discount policy. Fig. 5 depicts graphically the trade-o1s between the two incentives. The results indicate that there is a one-to-one non-linear relationship between t and d, even though the correlation coePcient between the two exceeds 0.9. However, the feasibility of such a tradeo1 ever being implemented by the wholesaler is highly questionable, since even a relatively small price discount of 10% would require, for the retailer to be indifferent, a pro2t-equivalent credit period of more than three
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Fig. 6. Constrained seller discount period.
months, too large an incentive period for most practical applications. Such result corresponds quite closely to what can be observed in the marketplace, which in turn provides a form of validation to the models of this paper. The last query refers to cases where the length of the promotional period, j ; j = DOP; DPP, may not be totally under the retailer’s control. Such an event normally occurs as a result of the vendor’s desire for a promotional period di1erent than the retailer’s optimal j or from the retailer’s perceived need to o1er the competition’s usual promotional period, as often happens to stores within, say, a shopping mall, involved in yet another common promotional e1ort to attract customers for all the stores in its midst. The retailer’s optimization model is now constrained by the size of j . The e1ect of such constraint is depicted in Fig. 6, which illustrates the di1erent e1ects of changes in j from 0 to 1, on pro2ts and on the decision variables. As expected, the e1ect on pro2ts is represented by a unimodal concave function with the maximum at the unconstrained optimal j . Of greater interest is its e1ect on the quantity variables and on the retailer’s depletion period. This e1ect is similar for the two policies, j = 1 (DPP); 2 (DOP), under consideration. The results indicate that (i) the portion of Qj sold by the
retailer at a discount increases along with j , to the point reaching the 100% level; consequently, (ii) the portion not sold at a discount decreases down to zero; but (iii) the magnitudes of the increases and decreases in Qj and qj , respectively, are such that the depletion period, Tj = j + qj =Rj , remains invariant with respect to the Juctuations in j , when qj ¿ 0, otherwise, Tj = j ; (iv) the e1ects on price and demand (not shown in Fig. 6) have the expected direction, i.e. j moves in the same (opposite) direction than Pj (Rj ); however, (v) consistent with earlier results and for similar reasons, the rate of change is substantially lower than for the quantities; and (vi) the larger the promotional period forced by the wholesaler, the lower is expected to be the amount of the discount passed on by the retailer on to its customers. The invariance of Tj when qj ¿ 0 has an analytic explanation. The constraint on j has the e1ect of eliminating one of the decision variables, say qj . Once the substitution of j for qj is made, simple arithmetic manipulations indicate that the 2rst-order optimality condition on Qj yields the following expression: When q2 ¿ 0; Tj = j + (Qj∗ − j Rj )=R0 for j = DPP; DOP
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= (P0 − c + d)=[(c − d)F + k] for j = DPP = (P0 − c − cFt)=(cF + k) for j = DOP:
(7)
It can be readily seen that the last expression for Tj , for qj ¿ 0 in (7) is a function exclusively of the model’s parameters and hence a constant. Further, the graphs on the quantities in Fig. 6 indicate that 2xing the length of the promotional period, j , beyond a certain point is equivalent to a retailer’s policy of passing on at least some of the discount, P0 − Pj 6 d, to all the units bought by the retailer in the trade promotion at a unit discount of d. In these instances, qj = 0 and Tj = j . The graph for Tj in Fig. 6, illustrates both cases. For the DPP option, q1 ¿ 0, for almost the entire range of values of 1 considered, up to 1 0:83. Hence, T1 is constant throughout, except for the last portion. For the DOP option, q2 = 0, beyond 2 ¿ 0:4 and thus T2 = 2 , beyond that point. 5. Some concluding comments Several managerial implications may be extracted from the modelling, the properties and the numerical example of this study. Given the proliferation of payment reduction schemes designed to induce larger orders, it is important for the retailer to be able to assess their monetary impact and to compare the pro2tability of these sale promotion tools, under realistic conditions linking marketing as well as operations management concerns. The integration of the purchasing, inventory and sale implications of the vendor’s o1er on the retailer’s pro2t forms an integral part of the study. At the modelling side, the 2rst linkage is the use of the price-dependent demand, so common in marketing but basically ignored in the inventory literature. Further, the model is valid for the general demand form de2ned in (1), which substantially enhances the range of applicability of the DOP=DPP comparisons. The second takes into consideration the well-known marketing fact e.g. [20,3,22] that the value of retailer pass-through on the retailer’s own customers varies widely. Our modelling of this evidence is through the use of (Pj ; qj ) as decision variables. Pj determines the portion of the discount to be passed on. An additional form of forward buying is related to Qj − qj , the portion of the retailer’s special o1er to be sold by the retailer to its own customers under a retail promotion. A third linkage consists of the introduction of the inventory component, ordering and holding costs, normal in the operations management literature. The fourth linkage is the incorporation of the DOP option, another popular sales promotion tool [4] into this structure, which in turn requires (i) a decomposition of inventory holding costs into physical and 2nancial, with di1erent cost drivers; and (ii) a common planning horizon for the comparison among models. Properties 2– 4 also provide important linkages to observed marketing practice, as they present evidence of the pro2tability of discount pass-throughs (P2) and of larger one-time-only purchases on
the part of the retailer (P3), together with the conditions under which the DPP option dominates (P4). Additional managerial implications are reJected in the numerical example which in turn provide some sort of empirical validation for the model of this paper. Included here are: (i) the pro2tability of price-induced demand changes on the retailer’s part; (ii) the larger variability of quantities rather than of prices to forestall or at least minimize adverse short-term competitive reactions, when the retailer is faced with changes in the special 2nancial incentives; (iii) the diPculty of trading o1 a DOP o1er for an equally bene2cial DPP counterpart; (iv) the positive (negative) relationship between the retailer’s pass-through rate and the length of the retailer’s (wholesaler’s) promotional period. It is clear from this study that there does not exist a one-to-one linear correspondence between DPP and DOP optimal policies. Hence, any possible tradeo1 between them implies di1erent order and pricing strategies, which are diPcult to determine without solving each problem individually. This adds substantially to the complexity of the decision process and suggests additional analysis to fully characterize such tradeo1. Scanbacks e.g. [20], whereby compensation is based on actual performance as measured by scanner data, are being used with increasing frequency by manufacturers to control retail pass-through. The extent to which these pay-for-performance trade deals may alter the incentive structure merits further investigation. Another issue of interest is the e1ect of changing a product’s demand on other potential products in a retailer’s assortment. Placing the model within a multi-product framework allows for the inclusion of loss leaders into the formulation and for the study of the complementary=substitutability e1ects of these promotion practices [3]. Another unresolved issue of managerial importance is whether the demand will be lower in the post-incentive period. The empirical evidence on this point is contradictory [3]. A resolution of this dispute will help to elucidate the nature of the post-discount demand e1ect, so that it can be incorporated into the model. Finally, problems related to uncertainty e.g. [32,21], such as stochastic demand, stochastic deal timing and the expectation of future deals, may also a1ect the comparability of the two options. The study of these and other issues justify additional research. Acknowledgements Financial assistance for the completion of this research from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. A major portion of this research has been carried out while the 2rst mentioned author was on sabbatical at the Departamento de GestiQon de Empresas, Universidad PQublica de Navarra under the Sabbatical Program of the Ministry of Education, Government of Spain. We are grateful for the funding from the DirecciQon General de InvestigaciQon y Desarrollo, Ministerio de
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EducaciQon y Ciencia (Grant No. SAB 95-0428) and for the supportive facilities at UPNA. Appendix A. Proofs of properties 2-- 4 Proof of Property 2. To prove P2, it suPces to show that there exists at least one feasible, even if non-optimal discount or a credit policy, with a pro2t of Bj (Q0∗ ; Pj ; ¿ 0), which exceeds that, Bj (Q0∗ ; P0∗ ; = 0), associated with the optimal policy if no portion of the discount is passed along. Take the j = 2, credit case. Using the de2nition of B2 from Table 1, one obtains the following inequality: B2 (Q0∗ ; P2 ; ¿ 0) − B2 (Q0∗ ; P0∗ ; = 0) = − Q0∗ { − (1=R0 − 1=R2 )[(K + cF)Q0∗ =2 + B0∗ ]}: (A.1) It can be readily seen that for the pro2t di1erence in (A.1) to be positive all that is required is that the term in { } be negative. That is always feasible because the term to be subtracted from is strictly positive and thus its value constitutes a positive upper value for and thus for the non-optimal ¿ 0 alternative to dominate its = 0 counterpart. A similar argument may be developed for the discount, j = 1 case. Proof of Property 3. Combining the de2nition of Q1∗ from Table 2 and that of B0∗ from Table 1, the expression for Q1∗ may be written as follows: Q1∗ = [R∗1 (P1∗ − c + d) − R0 (P0∗ − c)]= [(c − d)F + K] + Q0∗ (cF + K)=[(c − d)F + K: (A.2) The proof consists of showing that the numerator of the 2rst term of the RHS of (A.2) is positive and that the ratio being multiplied by Q0∗ in (A.2) is greater than 1. The latter is obvious, since d ¿ 0. As for the former, (3) implies that P1∗ ¡ P0∗ , which together with (1) leads to R∗1 ¿ R∗0 and to (P1∗ − c + d) − (P0∗ − c) = (P0∗ − − c + d) − (P0∗ − c) = − + d ¿ 0: A similar argument leads to the proof that
(A.3) Q2∗
¿ Q0∗ .
Proof of Property 4. The proof is by contradiction. Suppose that B1 6 B2 , whenever d ¿ cFt, Then, using the de2nitions of B1 and B2 from Table 1, the following inequalities follow: B1 − B2 = cFtQ − d{Q + (F=2)[(Q − q)2 =R1 + q2 =R0 ]} 6 0 ⇒ d=cFt 6 Q={Q + (F=2)[(Q − q)2 =R1 + q2 =R0 ]}: (A.4)
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The contradiction arises in the second inequality of (A.4). By assumption, the LHS term has a value greater than 1. But the denominator of the RHS is greater in value than its numerator. Thus, the ratio is less than 1. Hence, the contradiction.
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